![Page 1: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/1.jpg)
Triple IntegralsMath 212
Brian D. Fitzpatrick
Duke University
February 26, 2020
MATH
![Page 2: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/2.jpg)
Overview
Triple IntegralsMotivationIterated IntegralsNonrectangular Regions
![Page 3: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/3.jpg)
Triple IntegralsMotivation
Example
An object conforms to the shape of a solid W in R3.
P1
f (P1)=14 kg/m3
P2f (P2)=7 kg/m3
P3
f (P3)=9 kg/m3
P4 f (P4)=21 kg/m3
Suppose f ∈ C (R3) measures density (kg/m3) throughout W .
DefinitionThe triple integral of f on W is∫∫∫
Wf dV = mass of W (in kg)
![Page 4: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/4.jpg)
Triple IntegralsMotivation
Example
An object conforms to the shape of a solid W in R3.
P1
f (P1)=14 kg/m3
P2f (P2)=7 kg/m3
P3
f (P3)=9 kg/m3
P4 f (P4)=21 kg/m3
Suppose f ∈ C (R3) measures density (kg/m3) throughout W .
DefinitionThe triple integral of f on W is∫∫∫
Wf dV = mass of W (in kg)
![Page 5: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/5.jpg)
Triple IntegralsMotivation
Example
An object conforms to the shape of a solid W in R3.
P1
f (P1)=14 kg/m3
P2f (P2)=7 kg/m3
P3
f (P3)=9 kg/m3
P4 f (P4)=21 kg/m3
Suppose f ∈ C (R3) measures density (kg/m3) throughout W .
DefinitionThe triple integral of f on W is∫∫∫
Wf dV = mass of W (in kg)
![Page 6: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/6.jpg)
Triple IntegralsMotivation
Example
An object conforms to the shape of a solid W in R3.
P1
f (P1)=14 kg/m3
P2f (P2)=7 kg/m3
P3
f (P3)=9 kg/m3
P4 f (P4)=21 kg/m3
Suppose f ∈ C (R3) measures density (kg/m3) throughout W .
DefinitionThe triple integral of f on W is∫∫∫
Wf dV = mass of W (in kg)
![Page 7: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/7.jpg)
Triple IntegralsMotivation
Example
An object conforms to the shape of a solid W in R3.
P1
f (P1)=14 kg/m3
P2f (P2)=7 kg/m3
P3
f (P3)=9 kg/m3
P4 f (P4)=21 kg/m3
Suppose f ∈ C (R3) measures density (kg/m3) throughout W .
DefinitionThe triple integral of f on W is∫∫∫
Wf dV = mass of W (in kg)
![Page 8: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/8.jpg)
Triple IntegralsMotivation
Example
An object conforms to the shape of a solid W in R3.
P1
f (P1)=14 kg/m3
P2f (P2)=7 kg/m3
P3
f (P3)=9 kg/m3
P4 f (P4)=21 kg/m3
Suppose f ∈ C (R3) measures density (kg/m3) throughout W .
DefinitionThe triple integral of f on W is∫∫∫
Wf dV = mass of W (in kg)
![Page 9: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/9.jpg)
Triple IntegralsMotivation
Example
An object conforms to the shape of a solid W in R3.
P1
f (P1)=14 kg/m3
P2f (P2)=7 kg/m3
P3
f (P3)=9 kg/m3
P4 f (P4)=21 kg/m3
Suppose f ∈ C (R3) measures density (kg/m3) throughout W .
DefinitionThe triple integral of f on W is∫∫∫
Wf dV = mass of W (in kg)
![Page 10: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/10.jpg)
Triple IntegralsMotivation
ObservationTracking units allows us to interpret double integrals.∫∫∫
Wf
mass units
volume unit
dV
volume unit
=
mass of W
![Page 11: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/11.jpg)
Triple IntegralsMotivation
ObservationTracking units allows us to interpret double integrals.∫∫∫
Wf
mass units
volume unit
dV
volume unit
=
mass of W
![Page 12: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/12.jpg)
Triple IntegralsMotivation
ObservationTracking units allows us to interpret double integrals.∫∫∫
Wf
mass units
volume unit
dV
volume unit
=
mass of W
![Page 13: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/13.jpg)
Triple IntegralsMotivation
ObservationTracking units allows us to interpret double integrals.∫∫∫
Wf
mass units
volume unit
dV
volume unit
= mass of W
![Page 14: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/14.jpg)
Triple IntegralsIterated Integrals
QuestionHow can we calculate a triple integral?
AnswerUse iterated integrals!
![Page 15: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/15.jpg)
Triple IntegralsIterated Integrals
QuestionHow can we calculate a triple integral?
AnswerUse iterated integrals!
![Page 16: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/16.jpg)
Triple IntegralsIterated Integrals
Example
Suppose f = xy − z ◦C/m3 measures density throughout
W = [1, 3]× [3, 5]× [−2, 2]
The mass of W is∫∫∫W
f dV =
∫ 3
1
∫ 5
3
∫ 2
−2xy − z dz dy dx =
∫ 3
132 x dx
=
∫ 3
1
∫ 5
3xyz − 1
2z2∣∣∣∣z=2
z=−2dy dx = 16 x2
∣∣x=3
x=1
=
∫ 3
1
∫ 5
34 xy dy dx = 128 ◦C
=
∫ 3
12 xy2
∣∣y=5
y=3dx
![Page 17: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/17.jpg)
Triple IntegralsIterated Integrals
Example
Suppose f = xy − z ◦C/m3 measures density throughout
W = [1, 3]× [3, 5]× [−2, 2]
The mass of W is∫∫∫W
f dV =
∫ 3
1
∫ 5
3
∫ 2
−2xy − z dz dy dx =
∫ 3
132 x dx
=
∫ 3
1
∫ 5
3xyz − 1
2z2∣∣∣∣z=2
z=−2dy dx = 16 x2
∣∣x=3
x=1
=
∫ 3
1
∫ 5
34 xy dy dx = 128 ◦C
=
∫ 3
12 xy2
∣∣y=5
y=3dx
![Page 18: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/18.jpg)
Triple IntegralsIterated Integrals
Example
Suppose f = xy − z ◦C/m3 measures density throughout
W = [1, 3]× [3, 5]× [−2, 2]
The mass of W is∫∫∫W
f dV =
∫ 3
1
∫ 5
3
∫ 2
−2xy − z dz dy dx
=
∫ 3
132 x dx
=
∫ 3
1
∫ 5
3xyz − 1
2z2∣∣∣∣z=2
z=−2dy dx = 16 x2
∣∣x=3
x=1
=
∫ 3
1
∫ 5
34 xy dy dx = 128 ◦C
=
∫ 3
12 xy2
∣∣y=5
y=3dx
![Page 19: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/19.jpg)
Triple IntegralsIterated Integrals
Example
Suppose f = xy − z ◦C/m3 measures density throughout
W = [1, 3]× [3, 5]× [−2, 2]
The mass of W is∫∫∫W
f dV =
∫ 3
1
∫ 5
3
∫ 2
−2xy − z dz dy dx
=
∫ 3
132 x dx
=
∫ 3
1
∫ 5
3xyz − 1
2z2∣∣∣∣z=2
z=−2dy dx = 16 x2
∣∣x=3
x=1
=
∫ 3
1
∫ 5
34 xy dy dx = 128 ◦C
=
∫ 3
12 xy2
∣∣y=5
y=3dx
![Page 20: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/20.jpg)
Triple IntegralsIterated Integrals
Example
Suppose f = xy − z ◦C/m3 measures density throughout
W = [1, 3]× [3, 5]× [−2, 2]
The mass of W is∫∫∫W
f dV =
∫ 3
1
∫ 5
3
∫ 2
−2xy − z dz dy dx
=
∫ 3
132 x dx
=
∫ 3
1
∫ 5
3xyz − 1
2z2∣∣∣∣z=2
z=−2dy dx
= 16 x2∣∣x=3
x=1
=
∫ 3
1
∫ 5
34 xy dy dx = 128 ◦C
=
∫ 3
12 xy2
∣∣y=5
y=3dx
![Page 21: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/21.jpg)
Triple IntegralsIterated Integrals
Example
Suppose f = xy − z ◦C/m3 measures density throughout
W = [1, 3]× [3, 5]× [−2, 2]
The mass of W is∫∫∫W
f dV =
∫ 3
1
∫ 5
3
∫ 2
−2xy − z dz dy dx
=
∫ 3
132 x dx
=
∫ 3
1
∫ 5
3xyz − 1
2z2∣∣∣∣z=2
z=−2dy dx
= 16 x2∣∣x=3
x=1
=
∫ 3
1
∫ 5
34 xy dy dx
= 128 ◦C
=
∫ 3
12 xy2
∣∣y=5
y=3dx
![Page 22: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/22.jpg)
Triple IntegralsIterated Integrals
Example
Suppose f = xy − z ◦C/m3 measures density throughout
W = [1, 3]× [3, 5]× [−2, 2]
The mass of W is∫∫∫W
f dV =
∫ 3
1
∫ 5
3
∫ 2
−2xy − z dz dy dx
=
∫ 3
132 x dx
=
∫ 3
1
∫ 5
3xyz − 1
2z2∣∣∣∣z=2
z=−2dy dx
= 16 x2∣∣x=3
x=1
=
∫ 3
1
∫ 5
34 xy dy dx
= 128 ◦C
=
∫ 3
12 xy2
∣∣y=5
y=3dx
![Page 23: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/23.jpg)
Triple IntegralsIterated Integrals
Example
Suppose f = xy − z ◦C/m3 measures density throughout
W = [1, 3]× [3, 5]× [−2, 2]
The mass of W is∫∫∫W
f dV =
∫ 3
1
∫ 5
3
∫ 2
−2xy − z dz dy dx =
∫ 3
132 x dx
=
∫ 3
1
∫ 5
3xyz − 1
2z2∣∣∣∣z=2
z=−2dy dx
= 16 x2∣∣x=3
x=1
=
∫ 3
1
∫ 5
34 xy dy dx
= 128 ◦C
=
∫ 3
12 xy2
∣∣y=5
y=3dx
![Page 24: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/24.jpg)
Triple IntegralsIterated Integrals
Example
Suppose f = xy − z ◦C/m3 measures density throughout
W = [1, 3]× [3, 5]× [−2, 2]
The mass of W is∫∫∫W
f dV =
∫ 3
1
∫ 5
3
∫ 2
−2xy − z dz dy dx =
∫ 3
132 x dx
=
∫ 3
1
∫ 5
3xyz − 1
2z2∣∣∣∣z=2
z=−2dy dx =
16 x2∣∣x=3
x=1
=
∫ 3
1
∫ 5
34 xy dy dx
= 128 ◦C
=
∫ 3
12 xy2
∣∣y=5
y=3dx
![Page 25: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/25.jpg)
Triple IntegralsIterated Integrals
Example
Suppose f = xy − z ◦C/m3 measures density throughout
W = [1, 3]× [3, 5]× [−2, 2]
The mass of W is∫∫∫W
f dV =
∫ 3
1
∫ 5
3
∫ 2
−2xy − z dz dy dx =
∫ 3
132 x dx
=
∫ 3
1
∫ 5
3xyz − 1
2z2∣∣∣∣z=2
z=−2dy dx = 16 x2
∣∣x=3
x=1
=
∫ 3
1
∫ 5
34 xy dy dx =
128 ◦C
=
∫ 3
12 xy2
∣∣y=5
y=3dx
![Page 26: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/26.jpg)
Triple IntegralsIterated Integrals
Example
Suppose f = xy − z ◦C/m3 measures density throughout
W = [1, 3]× [3, 5]× [−2, 2]
The mass of W is∫∫∫W
f dV =
∫ 3
1
∫ 5
3
∫ 2
−2xy − z dz dy dx =
∫ 3
132 x dx
=
∫ 3
1
∫ 5
3xyz − 1
2z2∣∣∣∣z=2
z=−2dy dx = 16 x2
∣∣x=3
x=1
=
∫ 3
1
∫ 5
34 xy dy dx = 128 ◦C
=
∫ 3
12 xy2
∣∣y=5
y=3dx
![Page 27: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/27.jpg)
Triple IntegralsIterated Integrals
ObservationDouble integrals over rectangular regions come in two flavors
∫ x2
x1
∫ y2
y1
f dy dx
(x-slicing)
∫ y2
y1
∫ x2
x1
f dx dy
(y -slicing)
Triple integrals over rectangular regions come in six flavors∫ x2
x1
∫ y2
y1
∫ z2
z1
f dz dy dx
∫ x2
x1
∫ z2
z1
∫ y2
y1
f dy dz dx
(x-slicing)
∫ y2
y1
∫ x2
x1
∫ z2
z1
f dz dx dy
∫ y2
y1
∫ z2
z1
∫ x2
x1
f dx dz dy
(y -slicing)
∫ z2
z1
∫ x2
x1
∫ y2
y1
f dy dx dz
∫ z2
z1
∫ y2
y1
∫ x2
x1
f dx dy dz
(z-slicing)
![Page 28: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/28.jpg)
Triple IntegralsIterated Integrals
ObservationDouble integrals over rectangular regions come in two flavors∫ x2
x1
∫ y2
y1
f dy dx
(x-slicing)
∫ y2
y1
∫ x2
x1
f dx dy
(y -slicing)
Triple integrals over rectangular regions come in six flavors∫ x2
x1
∫ y2
y1
∫ z2
z1
f dz dy dx
∫ x2
x1
∫ z2
z1
∫ y2
y1
f dy dz dx
(x-slicing)
∫ y2
y1
∫ x2
x1
∫ z2
z1
f dz dx dy
∫ y2
y1
∫ z2
z1
∫ x2
x1
f dx dz dy
(y -slicing)
∫ z2
z1
∫ x2
x1
∫ y2
y1
f dy dx dz
∫ z2
z1
∫ y2
y1
∫ x2
x1
f dx dy dz
(z-slicing)
![Page 29: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/29.jpg)
Triple IntegralsIterated Integrals
ObservationDouble integrals over rectangular regions come in two flavors∫ x2
x1
∫ y2
y1
f dy dx
(x-slicing)
∫ y2
y1
∫ x2
x1
f dx dy
(y -slicing)
Triple integrals over rectangular regions come in six flavors∫ x2
x1
∫ y2
y1
∫ z2
z1
f dz dy dx
∫ x2
x1
∫ z2
z1
∫ y2
y1
f dy dz dx
(x-slicing)
∫ y2
y1
∫ x2
x1
∫ z2
z1
f dz dx dy
∫ y2
y1
∫ z2
z1
∫ x2
x1
f dx dz dy
(y -slicing)
∫ z2
z1
∫ x2
x1
∫ y2
y1
f dy dx dz
∫ z2
z1
∫ y2
y1
∫ x2
x1
f dx dy dz
(z-slicing)
![Page 30: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/30.jpg)
Triple IntegralsIterated Integrals
ObservationDouble integrals over rectangular regions come in two flavors∫ x2
x1
∫ y2
y1
f dy dx (x-slicing)
∫ y2
y1
∫ x2
x1
f dx dy
(y -slicing)
Triple integrals over rectangular regions come in six flavors∫ x2
x1
∫ y2
y1
∫ z2
z1
f dz dy dx
∫ x2
x1
∫ z2
z1
∫ y2
y1
f dy dz dx
(x-slicing)
∫ y2
y1
∫ x2
x1
∫ z2
z1
f dz dx dy
∫ y2
y1
∫ z2
z1
∫ x2
x1
f dx dz dy
(y -slicing)
∫ z2
z1
∫ x2
x1
∫ y2
y1
f dy dx dz
∫ z2
z1
∫ y2
y1
∫ x2
x1
f dx dy dz
(z-slicing)
![Page 31: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/31.jpg)
Triple IntegralsIterated Integrals
ObservationDouble integrals over rectangular regions come in two flavors∫ x2
x1
∫ y2
y1
f dy dx (x-slicing)
∫ y2
y1
∫ x2
x1
f dx dy
(y -slicing)
Triple integrals over rectangular regions come in six flavors∫ x2
x1
∫ y2
y1
∫ z2
z1
f dz dy dx
∫ x2
x1
∫ z2
z1
∫ y2
y1
f dy dz dx
(x-slicing)
∫ y2
y1
∫ x2
x1
∫ z2
z1
f dz dx dy
∫ y2
y1
∫ z2
z1
∫ x2
x1
f dx dz dy
(y -slicing)
∫ z2
z1
∫ x2
x1
∫ y2
y1
f dy dx dz
∫ z2
z1
∫ y2
y1
∫ x2
x1
f dx dy dz
(z-slicing)
![Page 32: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/32.jpg)
Triple IntegralsIterated Integrals
ObservationDouble integrals over rectangular regions come in two flavors∫ x2
x1
∫ y2
y1
f dy dx (x-slicing)
∫ y2
y1
∫ x2
x1
f dx dy (y -slicing)
Triple integrals over rectangular regions come in six flavors∫ x2
x1
∫ y2
y1
∫ z2
z1
f dz dy dx
∫ x2
x1
∫ z2
z1
∫ y2
y1
f dy dz dx
(x-slicing)
∫ y2
y1
∫ x2
x1
∫ z2
z1
f dz dx dy
∫ y2
y1
∫ z2
z1
∫ x2
x1
f dx dz dy
(y -slicing)
∫ z2
z1
∫ x2
x1
∫ y2
y1
f dy dx dz
∫ z2
z1
∫ y2
y1
∫ x2
x1
f dx dy dz
(z-slicing)
![Page 33: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/33.jpg)
Triple IntegralsIterated Integrals
ObservationDouble integrals over rectangular regions come in two flavors∫ x2
x1
∫ y2
y1
f dy dx (x-slicing)
∫ y2
y1
∫ x2
x1
f dx dy (y -slicing)
Triple integrals over rectangular regions come in six flavors
∫ x2
x1
∫ y2
y1
∫ z2
z1
f dz dy dx
∫ x2
x1
∫ z2
z1
∫ y2
y1
f dy dz dx
(x-slicing)
∫ y2
y1
∫ x2
x1
∫ z2
z1
f dz dx dy
∫ y2
y1
∫ z2
z1
∫ x2
x1
f dx dz dy
(y -slicing)
∫ z2
z1
∫ x2
x1
∫ y2
y1
f dy dx dz
∫ z2
z1
∫ y2
y1
∫ x2
x1
f dx dy dz
(z-slicing)
![Page 34: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/34.jpg)
Triple IntegralsIterated Integrals
ObservationDouble integrals over rectangular regions come in two flavors∫ x2
x1
∫ y2
y1
f dy dx (x-slicing)
∫ y2
y1
∫ x2
x1
f dx dy (y -slicing)
Triple integrals over rectangular regions come in six flavors∫ x2
x1
∫ y2
y1
∫ z2
z1
f dz dy dx
∫ x2
x1
∫ z2
z1
∫ y2
y1
f dy dz dx
(x-slicing)
∫ y2
y1
∫ x2
x1
∫ z2
z1
f dz dx dy
∫ y2
y1
∫ z2
z1
∫ x2
x1
f dx dz dy
(y -slicing)
∫ z2
z1
∫ x2
x1
∫ y2
y1
f dy dx dz
∫ z2
z1
∫ y2
y1
∫ x2
x1
f dx dy dz
(z-slicing)
![Page 35: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/35.jpg)
Triple IntegralsIterated Integrals
ObservationDouble integrals over rectangular regions come in two flavors∫ x2
x1
∫ y2
y1
f dy dx (x-slicing)
∫ y2
y1
∫ x2
x1
f dx dy (y -slicing)
Triple integrals over rectangular regions come in six flavors∫ x2
x1
∫ y2
y1
∫ z2
z1
f dz dy dx
∫ x2
x1
∫ z2
z1
∫ y2
y1
f dy dz dx
(x-slicing)
∫ y2
y1
∫ x2
x1
∫ z2
z1
f dz dx dy
∫ y2
y1
∫ z2
z1
∫ x2
x1
f dx dz dy
(y -slicing)
∫ z2
z1
∫ x2
x1
∫ y2
y1
f dy dx dz
∫ z2
z1
∫ y2
y1
∫ x2
x1
f dx dy dz
(z-slicing)
![Page 36: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/36.jpg)
Triple IntegralsIterated Integrals
ObservationDouble integrals over rectangular regions come in two flavors∫ x2
x1
∫ y2
y1
f dy dx (x-slicing)
∫ y2
y1
∫ x2
x1
f dx dy (y -slicing)
Triple integrals over rectangular regions come in six flavors∫ x2
x1
∫ y2
y1
∫ z2
z1
f dz dy dx
∫ x2
x1
∫ z2
z1
∫ y2
y1
f dy dz dx (x-slicing)∫ y2
y1
∫ x2
x1
∫ z2
z1
f dz dx dy
∫ y2
y1
∫ z2
z1
∫ x2
x1
f dx dz dy
(y -slicing)
∫ z2
z1
∫ x2
x1
∫ y2
y1
f dy dx dz
∫ z2
z1
∫ y2
y1
∫ x2
x1
f dx dy dz
(z-slicing)
![Page 37: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/37.jpg)
Triple IntegralsIterated Integrals
ObservationDouble integrals over rectangular regions come in two flavors∫ x2
x1
∫ y2
y1
f dy dx (x-slicing)
∫ y2
y1
∫ x2
x1
f dx dy (y -slicing)
Triple integrals over rectangular regions come in six flavors∫ x2
x1
∫ y2
y1
∫ z2
z1
f dz dy dx
∫ x2
x1
∫ z2
z1
∫ y2
y1
f dy dz dx (x-slicing)∫ y2
y1
∫ x2
x1
∫ z2
z1
f dz dx dy
∫ y2
y1
∫ z2
z1
∫ x2
x1
f dx dz dy
(y -slicing)
∫ z2
z1
∫ x2
x1
∫ y2
y1
f dy dx dz
∫ z2
z1
∫ y2
y1
∫ x2
x1
f dx dy dz
(z-slicing)
![Page 38: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/38.jpg)
Triple IntegralsIterated Integrals
ObservationDouble integrals over rectangular regions come in two flavors∫ x2
x1
∫ y2
y1
f dy dx (x-slicing)
∫ y2
y1
∫ x2
x1
f dx dy (y -slicing)
Triple integrals over rectangular regions come in six flavors∫ x2
x1
∫ y2
y1
∫ z2
z1
f dz dy dx
∫ x2
x1
∫ z2
z1
∫ y2
y1
f dy dz dx (x-slicing)∫ y2
y1
∫ x2
x1
∫ z2
z1
f dz dx dy
∫ y2
y1
∫ z2
z1
∫ x2
x1
f dx dz dy (y -slicing)∫ z2
z1
∫ x2
x1
∫ y2
y1
f dy dx dz
∫ z2
z1
∫ y2
y1
∫ x2
x1
f dx dy dz
(z-slicing)
![Page 39: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/39.jpg)
Triple IntegralsIterated Integrals
ObservationDouble integrals over rectangular regions come in two flavors∫ x2
x1
∫ y2
y1
f dy dx (x-slicing)
∫ y2
y1
∫ x2
x1
f dx dy (y -slicing)
Triple integrals over rectangular regions come in six flavors∫ x2
x1
∫ y2
y1
∫ z2
z1
f dz dy dx
∫ x2
x1
∫ z2
z1
∫ y2
y1
f dy dz dx (x-slicing)∫ y2
y1
∫ x2
x1
∫ z2
z1
f dz dx dy
∫ y2
y1
∫ z2
z1
∫ x2
x1
f dx dz dy (y -slicing)∫ z2
z1
∫ x2
x1
∫ y2
y1
f dy dx dz
∫ z2
z1
∫ y2
y1
∫ x2
x1
f dx dy dz
(z-slicing)
![Page 40: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/40.jpg)
Triple IntegralsIterated Integrals
ObservationDouble integrals over rectangular regions come in two flavors∫ x2
x1
∫ y2
y1
f dy dx (x-slicing)
∫ y2
y1
∫ x2
x1
f dx dy (y -slicing)
Triple integrals over rectangular regions come in six flavors∫ x2
x1
∫ y2
y1
∫ z2
z1
f dz dy dx
∫ x2
x1
∫ z2
z1
∫ y2
y1
f dy dz dx (x-slicing)∫ y2
y1
∫ x2
x1
∫ z2
z1
f dz dx dy
∫ y2
y1
∫ z2
z1
∫ x2
x1
f dx dz dy (y -slicing)∫ z2
z1
∫ x2
x1
∫ y2
y1
f dy dx dz
∫ z2
z1
∫ y2
y1
∫ x2
x1
f dx dy dz (z-slicing)
![Page 41: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/41.jpg)
Triple IntegralsNonrectangular Regions
QuestionHow do we compute
∫∫∫W f dV if W is not rectangular?
AnswerOur slicing method will depend on the shape of W .
![Page 42: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/42.jpg)
Triple IntegralsNonrectangular Regions
QuestionHow do we compute
∫∫∫W f dV if W is not rectangular?
AnswerOur slicing method will depend on the shape of W .
![Page 43: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/43.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
y
z
6−x2
6−x3
x + 2 y + 3 z = 6
Each x-slice leaves an imprint on the yz-plane.
∫∫∫W
f dV =
∫ 6
0
∫ 6−x2
0
∫ 6−x−2 y3
0f dz dy dx =
∫ 6
0
∫ 6−x3
0
∫ 6−x−3 z2
0f dy dz dx
![Page 44: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/44.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
y
z
6−x2
6−x3
x + 2 y + 3 z = 6
Each x-slice leaves an imprint on the yz-plane.
∫∫∫W
f dV =
∫ 6
0
∫ 6−x2
0
∫ 6−x−2 y3
0f dz dy dx =
∫ 6
0
∫ 6−x3
0
∫ 6−x−3 z2
0f dy dz dx
![Page 45: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/45.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
y
z
6−x2
6−x3
x + 2 y + 3 z = 6
Each x-slice leaves an imprint on the yz-plane.
∫∫∫W
f dV =
∫ 6
0
∫ 6−x2
0
∫ 6−x−2 y3
0f dz dy dx =
∫ 6
0
∫ 6−x3
0
∫ 6−x−3 z2
0f dy dz dx
![Page 46: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/46.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
y
z
6−x2
6−x3
x + 2 y + 3 z = 6
Each x-slice leaves an imprint on the yz-plane.
∫∫∫W
f dV =
∫ 6
0
∫ 6−x2
0
∫ 6−x−2 y3
0f dz dy dx =
∫ 6
0
∫ 6−x3
0
∫ 6−x−3 z2
0f dy dz dx
![Page 47: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/47.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
y
z
6−x2
6−x3
x + 2 y + 3 z = 6
Each x-slice leaves an imprint on the yz-plane.
∫∫∫W
f dV =
∫ 6
0
∫ 6−x2
0
∫ 6−x−2 y3
0f dz dy dx =
∫ 6
0
∫ 6−x3
0
∫ 6−x−3 z2
0f dy dz dx
![Page 48: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/48.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
y
z
6−x2
6−x3
x + 2 y + 3 z = 6
Each x-slice leaves an imprint on the yz-plane.
∫∫∫W
f dV =
∫ 6
0
∫ 6−x2
0
∫ 6−x−2 y3
0f dz dy dx =
∫ 6
0
∫ 6−x3
0
∫ 6−x−3 z2
0f dy dz dx
![Page 49: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/49.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
y
z
6−x2
6−x3
x + 2 y + 3 z = 6
Each x-slice leaves an imprint on the yz-plane.
∫∫∫W
f dV =
∫ 6
0
∫ 6−x2
0
∫ 6−x−2 y3
0f dz dy dx =
∫ 6
0
∫ 6−x3
0
∫ 6−x−3 z2
0f dy dz dx
![Page 50: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/50.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
y
z
6−x2
6−x3
x + 2 y + 3 z = 6
Each x-slice leaves an imprint on the yz-plane.
∫∫∫W
f dV =
∫ 6
0
∫ 6−x2
0
∫ 6−x−2 y3
0f dz dy dx =
∫ 6
0
∫ 6−x3
0
∫ 6−x−3 z2
0f dy dz dx
![Page 51: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/51.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
x
z
6− 2 y
6−2 y3
x + 2 y + 3 z = 6
Each y -slice leaves an imprint on the xz-plane.
∫∫∫W
f dV =
∫ 3
0
∫ 6−2 y
0
∫ 6−x−2 y3
0f dz dx dy =
∫ 3
0
∫ 6−2 y3
0
∫ 6−2 y−3 z
0f dx dz dy
![Page 52: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/52.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
x
z
6− 2 y
6−2 y3
x + 2 y + 3 z = 6
Each y -slice leaves an imprint on the xz-plane.
∫∫∫W
f dV =
∫ 3
0
∫ 6−2 y
0
∫ 6−x−2 y3
0f dz dx dy =
∫ 3
0
∫ 6−2 y3
0
∫ 6−2 y−3 z
0f dx dz dy
![Page 53: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/53.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
x
z
6− 2 y
6−2 y3
x + 2 y + 3 z = 6
Each y -slice leaves an imprint on the xz-plane.
∫∫∫W
f dV =
∫ 3
0
∫ 6−2 y
0
∫ 6−x−2 y3
0f dz dx dy =
∫ 3
0
∫ 6−2 y3
0
∫ 6−2 y−3 z
0f dx dz dy
![Page 54: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/54.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
x
z
6− 2 y
6−2 y3
x + 2 y + 3 z = 6
Each y -slice leaves an imprint on the xz-plane.
∫∫∫W
f dV =
∫ 3
0
∫ 6−2 y
0
∫ 6−x−2 y3
0f dz dx dy =
∫ 3
0
∫ 6−2 y3
0
∫ 6−2 y−3 z
0f dx dz dy
![Page 55: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/55.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
x
z
6− 2 y
6−2 y3
x + 2 y + 3 z = 6
Each y -slice leaves an imprint on the xz-plane.
∫∫∫W
f dV =
∫ 3
0
∫ 6−2 y
0
∫ 6−x−2 y3
0f dz dx dy =
∫ 3
0
∫ 6−2 y3
0
∫ 6−2 y−3 z
0f dx dz dy
![Page 56: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/56.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
x
z
6− 2 y
6−2 y3
x + 2 y + 3 z = 6
Each y -slice leaves an imprint on the xz-plane.
∫∫∫W
f dV =
∫ 3
0
∫ 6−2 y
0
∫ 6−x−2 y3
0f dz dx dy =
∫ 3
0
∫ 6−2 y3
0
∫ 6−2 y−3 z
0f dx dz dy
![Page 57: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/57.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
x
z
6− 2 y
6−2 y3
x + 2 y + 3 z = 6
Each y -slice leaves an imprint on the xz-plane.
∫∫∫W
f dV =
∫ 3
0
∫ 6−2 y
0
∫ 6−x−2 y3
0f dz dx dy =
∫ 3
0
∫ 6−2 y3
0
∫ 6−2 y−3 z
0f dx dz dy
![Page 58: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/58.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
x
z
6− 2 y
6−2 y3
x + 2 y + 3 z = 6
Each y -slice leaves an imprint on the xz-plane.
∫∫∫W
f dV =
∫ 3
0
∫ 6−2 y
0
∫ 6−x−2 y3
0f dz dx dy =
∫ 3
0
∫ 6−2 y3
0
∫ 6−2 y−3 z
0f dx dz dy
![Page 59: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/59.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
x
y
6− 3 z
6−3 z2
x + 2 y + 3 z = 6
Each z-slice leaves an imprint on the xy -plane.
∫∫∫W
f dV =
∫ 2
0
∫ 6−3 z
0
∫ 6−x−3 z2
0f dy dx dz =
∫ 2
0
∫ 6−3 z2
0
∫ 6−2 y−3 z
0f dx dy dz
![Page 60: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/60.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
x
y
6− 3 z
6−3 z2
x + 2 y + 3 z = 6
Each z-slice leaves an imprint on the xy -plane.
∫∫∫W
f dV =
∫ 2
0
∫ 6−3 z
0
∫ 6−x−3 z2
0f dy dx dz =
∫ 2
0
∫ 6−3 z2
0
∫ 6−2 y−3 z
0f dx dy dz
![Page 61: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/61.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
x
y
6− 3 z
6−3 z2
x + 2 y + 3 z = 6
Each z-slice leaves an imprint on the xy -plane.
∫∫∫W
f dV =
∫ 2
0
∫ 6−3 z
0
∫ 6−x−3 z2
0f dy dx dz =
∫ 2
0
∫ 6−3 z2
0
∫ 6−2 y−3 z
0f dx dy dz
![Page 62: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/62.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
x
y
6− 3 z
6−3 z2
x + 2 y + 3 z = 6
Each z-slice leaves an imprint on the xy -plane.
∫∫∫W
f dV =
∫ 2
0
∫ 6−3 z
0
∫ 6−x−3 z2
0f dy dx dz =
∫ 2
0
∫ 6−3 z2
0
∫ 6−2 y−3 z
0f dx dy dz
![Page 63: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/63.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
x
y
6− 3 z
6−3 z2
x + 2 y + 3 z = 6
Each z-slice leaves an imprint on the xy -plane.
∫∫∫W
f dV =
∫ 2
0
∫ 6−3 z
0
∫ 6−x−3 z2
0f dy dx dz =
∫ 2
0
∫ 6−3 z2
0
∫ 6−2 y−3 z
0f dx dy dz
![Page 64: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/64.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
x
y
6− 3 z
6−3 z2
x + 2 y + 3 z = 6
Each z-slice leaves an imprint on the xy -plane.
∫∫∫W
f dV =
∫ 2
0
∫ 6−3 z
0
∫ 6−x−3 z2
0f dy dx dz =
∫ 2
0
∫ 6−3 z2
0
∫ 6−2 y−3 z
0f dx dy dz
![Page 65: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/65.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
x
y
6− 3 z
6−3 z2
x + 2 y + 3 z = 6
Each z-slice leaves an imprint on the xy -plane.
∫∫∫W
f dV =
∫ 2
0
∫ 6−3 z
0
∫ 6−x−3 z2
0f dy dx dz =
∫ 2
0
∫ 6−3 z2
0
∫ 6−2 y−3 z
0f dx dy dz
![Page 66: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/66.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x + 2 y + 3 z ≤ 6.
x
y
z
63
2
x
y
6− 3 z
6−3 z2
x + 2 y + 3 z = 6
Each z-slice leaves an imprint on the xy -plane.
∫∫∫W
f dV =
∫ 2
0
∫ 6−3 z
0
∫ 6−x−3 z2
0f dy dx dz =
∫ 2
0
∫ 6−3 z2
0
∫ 6−2 y−3 z
0f dx dy dz
![Page 67: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/67.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x2 + y2 ≤ z ≤ 1.
x
y
z
y
z
√1− x2
x2
1
z = x2 + y2
Each x-slice leaves an imprint on the yz-plane.
∫∫∫W
f dV =
∫ 1
0
∫ √1−x20
∫ 1
x2+y2
f dz dy dx =
∫ 1
0
∫ 1
x2
∫ √z−x20
f dy dz dx
![Page 68: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/68.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x2 + y2 ≤ z ≤ 1.
x
y
z
y
z
√1− x2
x2
1
z = x2 + y2
Each x-slice leaves an imprint on the yz-plane.
∫∫∫W
f dV =
∫ 1
0
∫ √1−x20
∫ 1
x2+y2
f dz dy dx =
∫ 1
0
∫ 1
x2
∫ √z−x20
f dy dz dx
![Page 69: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/69.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x2 + y2 ≤ z ≤ 1.
x
y
z
y
z
√1− x2
x2
1
z = x2 + y2
Each x-slice leaves an imprint on the yz-plane.
∫∫∫W
f dV =
∫ 1
0
∫ √1−x20
∫ 1
x2+y2
f dz dy dx =
∫ 1
0
∫ 1
x2
∫ √z−x20
f dy dz dx
![Page 70: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/70.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x2 + y2 ≤ z ≤ 1.
x
y
z
y
z
√1− x2
x2
1
z = x2 + y2
Each x-slice leaves an imprint on the yz-plane.
∫∫∫W
f dV =
∫ 1
0
∫ √1−x20
∫ 1
x2+y2
f dz dy dx =
∫ 1
0
∫ 1
x2
∫ √z−x20
f dy dz dx
![Page 71: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/71.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x2 + y2 ≤ z ≤ 1.
x
y
z
y
z
√1− x2
x2
1
z = x2 + y2
Each x-slice leaves an imprint on the yz-plane.
∫∫∫W
f dV =
∫ 1
0
∫ √1−x20
∫ 1
x2+y2
f dz dy dx =
∫ 1
0
∫ 1
x2
∫ √z−x20
f dy dz dx
![Page 72: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/72.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x2 + y2 ≤ z ≤ 1.
x
y
z
y
z
√1− x2
x2
1
z = x2 + y2
Each x-slice leaves an imprint on the yz-plane.∫∫∫W
f dV =
∫ 1
0
∫ √1−x20
∫ 1
x2+y2
f dz dy dx =
∫ 1
0
∫ 1
x2
∫ √z−x20
f dy dz dx
![Page 73: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/73.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x2 + y2 ≤ z ≤ 1.
x
y
z
y
z
√1− x2
x2
1
z = x2 + y2
Each x-slice leaves an imprint on the yz-plane.∫∫∫W
f dV =
∫ 1
0
∫ √1−x20
∫ 1
x2+y2
f dz dy dx =
∫ 1
0
∫ 1
x2
∫ √z−x20
f dy dz dx
![Page 74: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/74.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x2 + y2 ≤ z ≤ 1.
x
y
z
y
z
√1− x2
x2
1
z = x2 + y2
Each x-slice leaves an imprint on the yz-plane.∫∫∫W
f dV =
∫ 1
0
∫ √1−x20
∫ 1
x2+y2
f dz dy dx =
∫ 1
0
∫ 1
x2
∫ √z−x20
f dy dz dx
![Page 75: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/75.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x2 + y2 ≤ z ≤ 1.
x
y
z
x
y
√z
√z
z = x2 + y2
Each z-slice leaves an imprint on the xy -plane.
∫∫∫W
f dV =
∫ 1
0
∫ √z0
∫ √z−x20
f dy dx dz =
∫ 1
0
∫ √z0
∫ √z−y2
0f dx dy dz
![Page 76: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/76.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x2 + y2 ≤ z ≤ 1.
x
y
z
x
y
√z
√z
z = x2 + y2
Each z-slice leaves an imprint on the xy -plane.
∫∫∫W
f dV =
∫ 1
0
∫ √z0
∫ √z−x20
f dy dx dz =
∫ 1
0
∫ √z0
∫ √z−y2
0f dx dy dz
![Page 77: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/77.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x2 + y2 ≤ z ≤ 1.
x
y
z
x
y
√z
√z
z = x2 + y2
Each z-slice leaves an imprint on the xy -plane.
∫∫∫W
f dV =
∫ 1
0
∫ √z0
∫ √z−x20
f dy dx dz =
∫ 1
0
∫ √z0
∫ √z−y2
0f dx dy dz
![Page 78: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/78.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x2 + y2 ≤ z ≤ 1.
x
y
z
x
y
√z
√z
z = x2 + y2
Each z-slice leaves an imprint on the xy -plane.
∫∫∫W
f dV =
∫ 1
0
∫ √z0
∫ √z−x20
f dy dx dz =
∫ 1
0
∫ √z0
∫ √z−y2
0f dx dy dz
![Page 79: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/79.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x2 + y2 ≤ z ≤ 1.
x
y
z
x
y
√z
√z
z = x2 + y2
Each z-slice leaves an imprint on the xy -plane.
∫∫∫W
f dV =
∫ 1
0
∫ √z0
∫ √z−x20
f dy dx dz =
∫ 1
0
∫ √z0
∫ √z−y2
0f dx dy dz
![Page 80: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/80.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x2 + y2 ≤ z ≤ 1.
x
y
z
x
y
√z
√z
z = x2 + y2
Each z-slice leaves an imprint on the xy -plane.
∫∫∫W
f dV =
∫ 1
0
∫ √z0
∫ √z−x20
f dy dx dz =
∫ 1
0
∫ √z0
∫ √z−y2
0f dx dy dz
![Page 81: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/81.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x2 + y2 ≤ z ≤ 1.
x
y
z
x
y
√z
√z
z = x2 + y2
Each z-slice leaves an imprint on the xy -plane.
∫∫∫W
f dV =
∫ 1
0
∫ √z0
∫ √z−x20
f dy dx dz =
∫ 1
0
∫ √z0
∫ √z−y2
0f dx dy dz
![Page 82: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/82.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the “first octant” part of x2 + y2 ≤ z ≤ 1.
x
y
z
x
y
√z
√z
z = x2 + y2
Each z-slice leaves an imprint on the xy -plane.
∫∫∫W
f dV =
∫ 1
0
∫ √z0
∫ √z−x20
f dy dx dz =
∫ 1
0
∫ √z0
∫ √z−y2
0f dx dy dz
![Page 83: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/83.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the region 2 x2 + 2 z2 ≤ y ≤ x2 + z2 + 4.
y
y = x2 + z2 + 4
y = 2 x2 + 2 z2
x2 + z2 = 4
x
y
y = x2 + z2 + 4
y = 2 x2 + 2 z2
√4− z2−
√4− z2
Each z-slice leaves an imprint on the xy -plane.∫∫∫W
f dV =
∫ 2
−2
∫ √4−z2−√4−z2
∫ x2+z2+4
2 x2+2 z2f dy dx dz
![Page 84: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/84.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the region 2 x2 + 2 z2 ≤ y ≤ x2 + z2 + 4.
y
y = x2 + z2 + 4
y = 2 x2 + 2 z2
x2 + z2 = 4
x
y
y = x2 + z2 + 4
y = 2 x2 + 2 z2
√4− z2−
√4− z2
Each z-slice leaves an imprint on the xy -plane.∫∫∫W
f dV =
∫ 2
−2
∫ √4−z2−√4−z2
∫ x2+z2+4
2 x2+2 z2f dy dx dz
![Page 85: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/85.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the region 2 x2 + 2 z2 ≤ y ≤ x2 + z2 + 4.
y
y = x2 + z2 + 4
y = 2 x2 + 2 z2
x2 + z2 = 4
x
y
y = x2 + z2 + 4
y = 2 x2 + 2 z2
√4− z2−
√4− z2
Each z-slice leaves an imprint on the xy -plane.∫∫∫W
f dV =
∫ 2
−2
∫ √4−z2−√4−z2
∫ x2+z2+4
2 x2+2 z2f dy dx dz
![Page 86: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/86.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the region 2 x2 + 2 z2 ≤ y ≤ x2 + z2 + 4.
y
y = x2 + z2 + 4
y = 2 x2 + 2 z2
x2 + z2 = 4
x
y
y = x2 + z2 + 4
y = 2 x2 + 2 z2
√4− z2−
√4− z2
Each z-slice leaves an imprint on the xy -plane.∫∫∫W
f dV =
∫ 2
−2
∫ √4−z2−√4−z2
∫ x2+z2+4
2 x2+2 z2f dy dx dz
![Page 87: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/87.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the region 2 x2 + 2 z2 ≤ y ≤ x2 + z2 + 4.
y
y = x2 + z2 + 4
y = 2 x2 + 2 z2
x2 + z2 = 4
x
y
y = x2 + z2 + 4
y = 2 x2 + 2 z2
√4− z2−
√4− z2
Each z-slice leaves an imprint on the xy -plane.
∫∫∫W
f dV =
∫ 2
−2
∫ √4−z2−√4−z2
∫ x2+z2+4
2 x2+2 z2f dy dx dz
![Page 88: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/88.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the region 2 x2 + 2 z2 ≤ y ≤ x2 + z2 + 4.
y
y = x2 + z2 + 4
y = 2 x2 + 2 z2
x2 + z2 = 4
x
y
y = x2 + z2 + 4
y = 2 x2 + 2 z2
√4− z2−
√4− z2
Each z-slice leaves an imprint on the xy -plane.
∫∫∫W
f dV =
∫ 2
−2
∫ √4−z2−√4−z2
∫ x2+z2+4
2 x2+2 z2f dy dx dz
![Page 89: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/89.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the region 2 x2 + 2 z2 ≤ y ≤ x2 + z2 + 4.
y
y = x2 + z2 + 4
y = 2 x2 + 2 z2
x2 + z2 = 4
x
y
y = x2 + z2 + 4
y = 2 x2 + 2 z2
√4− z2−
√4− z2
Each z-slice leaves an imprint on the xy -plane.∫∫∫W
f dV =
∫ 2
−2
∫ √4−z2−√4−z2
∫ x2+z2+4
2 x2+2 z2f dy dx dz
![Page 90: Triple Integrals - Math 212Triple Integrals Motivation Example An object conforms to the shape of a solid W in R3. P 1 f(P1)=14kg=m3 f(P2)=7kg=m3 P 2 P 3 f(P3)=9kg=m3 P 4 f(P4)=21kg=m3](https://reader030.vdocuments.site/reader030/viewer/2022041101/5edb01ba09ac2c67fa68a890/html5/thumbnails/90.jpg)
Triple IntegralsNonrectangular Regions
Example
Consider the region 2 x2 + 2 z2 ≤ y ≤ x2 + z2 + 4.
y
y = x2 + z2 + 4
y = 2 x2 + 2 z2
x2 + z2 = 4
x
y
y = x2 + z2 + 4
y = 2 x2 + 2 z2
√4− z2−
√4− z2
Each z-slice leaves an imprint on the xy -plane.∫∫∫W
f dV =
∫ 2
−2
∫ √4−z2−√4−z2
∫ x2+z2+4
2 x2+2 z2f dy dx dz