16.1 double integrals - linda green
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§16.1 DOUBLE INTEGRALS
§16.1 Double Integrals
After completing this section, students should be able to:
• Give the definition of a double integral in terms of the limit of a Riemann sum.
• Approximate a double integral using a Riemann sum if given information about afunction’s values at various points, either from a contour map or a table of valuesor an explicit function.
• Use the idea that a double integral represents volume to compute some particularintegrals whose integrand functions trace out spheres or cylinders from knowngeometry formulas.
• Compute the integral of a function of two variables over a rectangular region byevaluating iterated integrals (i.e. integrating first in the x direction and then in they direction or vice versa).
• Define and compute the average value of a function of two variables over a region.
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§16.1 DOUBLE INTEGRALS
In Calculus 1, we defined the integral of f (x) over an interval [a, b] as the limit of aRiemann sum:
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§16.1 DOUBLE INTEGRALS
For a function f (x, y) over a rectangular region R = [a, b] ⇥ [c, d] we can define anintegral similarly:
ZZ
R
f (x, y)dA =
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§16.1 DOUBLE INTEGRALS
Example. Use a Riemann sum with m = 2 and n = 2 to estimate the value ofZ Z
R
xe�xy
dA, where R is the rectangle [0, 2] ⇥ [0, 1] ...
(a) using sample points in the upper right corners.
(b) using the Midpoint Rule.
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§16.1 DOUBLE INTEGRALS
Iterated Integrals
If R = [a, b] ⇥ [c, d], then
Volume =R
b
aA(x) dx , where A(x) =
SoRR
Rf (x, y) dA = .
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§16.1 DOUBLE INTEGRALS
Similarly, if we look at cross-sectional area in the other direction A(y), we can computethe volume over the rectangle R = [a, b] ⇥ [c, d] as:
RRR
f (x, y) dA = .
Theorem. Fubini’s Theorem If f is continuous on the rectangle R = [a, b] ⇥ [c, d], thenthe double integral equals the iterated integrals, i.e.RR
Rf (x, y) dA =
This theorem still holds true even if f is not continuous, as long as
•••
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§16.1 DOUBLE INTEGRALS
Example. Use Fubini’s Theorem to calculateZ Z
R
xe�xy
dA, where R = [0, 2] ⇥ [0, 1].
END OF VIDEOS
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§16.1 DOUBLE INTEGRALS
Example. For each problem, estimate the integral of f (x, y) over the rectangle specified:
1. The figure shows the level curves of a function f . EstimateRR
Rf (x, y) dA where
R = [0, 2] ⇥ [0, 2].
2. EstimateRR
Rf (x, y) dA, where R = [0, 4] ⇥ [2, 4], using the midpoint rule with
n = m = 2.
3. FindZZ
R
q9 � y2 dA, where R = [0, 8] ⇥ [0, 3]. Hint: this integral represents the
volume of a familiar solid.
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§16.1 DOUBLE INTEGRALS
Example. CalculateZZ
R
(6x2y � 2x) dA, where
R = {(x, y)|1 x 4, 0 y 2}
Find the average value of f (x, y) = 6x2y � 2x over this same region.
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§16.1 DOUBLE INTEGRALS
Example. Find the volume enclosed by the surfacez = 1 + e
x sin y and the planes x = ±1, y = 0, y = ⇡, and z = 0.
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§16.1 DOUBLE INTEGRALS
Question. True or False:Z 6
1f (x)g(x) dx =
Z 6
1f (x) dx ·
Z 6
1g(x) dx
Question. True or False:Z 6
1
Z 9
5f (x)g(y) dx dy =
Z 9
5f (x) dx ·
Z 6
1g(y) dy
Question. True or False:Z Z
D
1 dA = area(D)
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§16.1 DOUBLE INTEGRALS
Example. When converted to an iterated integral, the following double integral iseasier to evaluate in one order than in the other. Choose the best order and evaluatethe integral.Z Z
R
4x3e
x2
ydA, where 0 < x 2 and 0 y 1.
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§16.1 DOUBLE INTEGRALS
Example. Use symmetry to evaluate the integrals.
(a)Z 3
�3
Z 5
0y sin(x2 + y
2) + sin(xy) dx dy
(b)Z 4
0
Z 4
0(x � y) cos(x � y) dx dy
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§16.1 DOUBLE INTEGRALS
Extra Example. Find an upper and lower bound forZZ
D
e�(x2+y
2)dx dy
where D is the disk {(x, y)|x2 + y2 1
4}
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§16.2 DOUBLE INTEGRALS OVER GENERAL REGIONS
§16.2 Double Integrals over General Regions
After completing this section, students should be able to:
• Determine if an integral is easier to compute dx then dy vs. dy then dx, based onthe shape of the region.
• Compute integrals over Type I and Type II regions.
• Change the order of integration to make an integral easier to compute.
• Break up a region into a union of Type I and Type II regions in order to computean integral as a sum of several integrals.
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§16.2 DOUBLE INTEGRALS OVER GENERAL REGIONS
Example. CalculateZZ
D
(x2 + 2y) dA for the region D bounded by the parabolas y = 2x2
and y = 1 + x2.
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§16.2 DOUBLE INTEGRALS OVER GENERAL REGIONS
Example. CalculateZZ
D
y dA for the region D bounded by the line y = x � 1 and the
parabola y2 = 2x + 6.
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§16.2 DOUBLE INTEGRALS OVER GENERAL REGIONS
These two regions are examples of Type I and Type II regions.
Type I Region Type II Region
For a Type I region,ZZ
D
f (x, y) dA =
For a Type II region,ZZ
D
f (x, y) dA =
END OF VIDEOS
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§16.2 DOUBLE INTEGRALS OVER GENERAL REGIONS
Review:
Type I Region Type II RegionFor a Type I region,
ZZ
D
f (x, y) dA =
For a Type II region,ZZ
D
f (x, y) dA =
Note. If a region is both a Type I region and a Type II region, sometimes, it is easier toevaluate the integral in one way instead of the other.
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§16.2 DOUBLE INTEGRALS OVER GENERAL REGIONS
Example. EvaluateZ Z
D
x2 + 2y dA where D is the region bounded by y = x, y = x
2,
x � 0
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§16.2 DOUBLE INTEGRALS OVER GENERAL REGIONS
Example. Find the volume of the region bounded by the paraboloid z = x2+ y
2+1 andthe planes x = 0, y = 0, z = 0, and x + y = 2.
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§16.2 DOUBLE INTEGRALS OVER GENERAL REGIONS
Example. Evaluate
Z 8
0
Z 2
3py
ex
4dx dy
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§16.2 DOUBLE INTEGRALS OVER GENERAL REGIONS
Question. True or False:Z 6
1
Z 9
5f (x)g(y) dx dy =
Z 9
5f (x) dx ·
Z 6
1g(y) dy
Question. True or False:Z 6
1
Zy
2
5f (x)g(y) dx dy =
Zy
2
5f (x) dx ·
Z 6
1g(y) dy
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§16.3 INTEGRATION USING POLAR COORDINATES
§16.3 Integration using Polar Coordinates
After completing this section, students should be able to:
• Recognize what types of integrals may be easier to compute using polar coordi-nates.
• Set up and compute an integral using polar coordinates.
• Convert an integral from Cartesian coordinates to polar coordinates.
• Give an informal justification of why dV is not given by just dr d✓ in polar coordi-nates, but requires an extra factor.
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§16.3 INTEGRATION USING POLAR COORDINATES
Example. EvaluateZZ
D
x2y dA, where D is the top half of the disk with center the
origin and radius 5.
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§16.3 INTEGRATION USING POLAR COORDINATES
Theorem. If f (x, y) is continuous on a polar rectangle R = {(r,✓)|↵ ✓ �, a r b},then
ZZ
R
f (x, y) dA =
Proof. (Where does the extra r come from?)
END OF VIDEO
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§16.3 INTEGRATION USING POLAR COORDINATES
Question. Which of the following representsZZ
D
(2x � y) dA, where
D = {(x, y) | 1 x2 + y
2 4 and 0 y x} ?
A.Z ⇡/4
✓=0
Z 2
r=1(2 cos✓ � sin✓) r dr d✓
B.Z ⇡/4
✓=0
Z 2
r=1(2 cos✓ � sin✓) r
2dr d✓
C.Z p4�y2
x=p
1�y2
Zx
y=0(2x � y) dy dx
Evaluate the integral.
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§16.3 INTEGRATION USING POLAR COORDINATES
Example. Find the volume enclosed by the top half of the hyperboloid �x2� y
2+z2 = 1
(with z > 0) and the sphere x2 + y
2 + z2 = 5.
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§16.3 INTEGRATION USING POLAR COORDINATES
Integration over more general regions:
Theorem. If f (x, y) is continuous on the region R = {(r,✓)|↵ ✓ �, h1(✓) r h2(✓)},then
ZZ
R
f (x, y) dA =
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§16.3 INTEGRATION USING POLAR COORDINATES
Example. Find the area of the region that is inside the cardioid r = 1+cos✓ and outsidethe circle
⇣x � 3
2
⌘2+ y
2 = 94.
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§16.3 INTEGRATION USING POLAR COORDINATES
Extra Example. Evaluate the integral by changing to polar coordinates:
Z 2
0
Z p4�x2
0e�x
2�y2
dy dx
A.R 2
0
R p4�r2 cos✓
0 e�r
2r dr d✓
B.R ⇡
0
R 20 e�r
2dr d✓
C.R ⇡/2�⇡/2R 2
0 e�r
2r dr d✓
D.R ⇡/2
0
R 20 e�r
2r dr d✓
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§16.3 INTEGRATION USING POLAR COORDINATES
Extra Example. The equation of the standard normal curve (with mean 0 and standarddeviation 1) is
f (x) =1p2⇡
e�x
22
Prove that the area under the standard normal curve is 1.
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§16.4 TRIPLE INTEGRALS
§16.4 Triple Integrals
After completing this section, students should be able to:
• Set up triple integrals to calculate volume.
• Change the order of integration for a triple integral.
• Calculate triple integrals by integrating one variable at a time.
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§16.4 TRIPLE INTEGRALS
The integral of f (x, y, z) over a rectangular box B = [a, b] ⇥ [c, d] ⇥ [r, s] can be definedas a limit of a Riemann sum:
ZZZ
B
f (x, y, z) dV =
Theorem. (Fubini’s Theorem for Triple Integrals) If f (x, y, z) is continuous over the boxB, then
ZZZ
B
f (x, y, z) dV =
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§16.4 TRIPLE INTEGRALS
Integrals over general regions.
Example. Evaluate the triple integral:
ZZZ
E
ez/y
dV
where E = {(x, , y, z)|0 y 1, y x 1, 0 z xy}
END OF VIDEO
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§16.4 TRIPLE INTEGRALS
Example. Set up the bounds of integration forZZZ
E
z dV, where E is bounded by the
cylinder y2 + z
2 = 9 and the planes x = 0, y = 3x, and z = 0 in the first octant.
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§16.4 TRIPLE INTEGRALS
Question. Suppose I want to calculate the area between the curves y = x and y = x2?
• Could I compute it with a single integral?
• Could I compute it with a double integral?
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§16.4 TRIPLE INTEGRALS
Example. Suppose I wanted to find the volume of this solid.
Could I compute it using a double integral?
Could I compute it using a triple integral?
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§16.4 TRIPLE INTEGRALS
Extra Example. Rewrite the integral by changing the order of integration in as manyways as possible.
Z 1
0
Z 1�x
0
Z 1�x2
0f (x, y, z) dz dy dx
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§16.4 TRIPLE INTEGRALS
Extra Example. Set up the integral to find the mass of the region E bounded by theparabolic cylinder z = 1 � y
2 and the planes x + z = 1, x = 0, and z = 0, given thedensity function ⇢(x, y, z) = 4.
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§16.4 TRIPLE INTEGRALS
Tips for setting up triple integrals:
• All else equal, integrate first in the ...
• To find the base ....
• To find the boundary curves of the base region ...
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
§16.5 Triple Integrals in Cylindrical and Spherical Coordinates
After completing this sections, students should be able to:
• Convert points and equations between Cartesian coordinates and cylindrical co-ordinates and spherical coordinates.
• Sketch simple regions given in cylindrical coordinates and spherical coordinates.
• Set up integrals triple integrals in cylindrical coordinates and spherical coordinates,given a description of the region to be integrated over and the equation for thefunction to integrate, where the equation and descriptions themselves may begiven in terms of Cartesian or cylindrical coordinates or spherical coordinates.
• Compute volumes and masses of solids using cylindrical coordinates and sphericalcoordinates.
• Recognize whether an integral is easier to compute using spherical, cylindrical, orCartesian coordinates.
• Convert integrals from Cartesian coordinates to cylindrical coordinates and spher-ical coordinates.
• Compute integrals in cylindrical coordinates and spherical coordinates.
• Give an informal justification of why dV is not given by just d⇢ d✓ d� in sphericalcoordinates, but requires an extra factor.
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
The cylindrical coordinates of a point P in space are given by (r,✓, z)
where z is ...
and r and ✓ are ...
As with polar coordinates, r can be positive or negative.
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Example. .
Note. Cartesian coordinates and cylindrical coordinates are related by:
• x =
• y =
• z =
• r2 =
• tan✓ =
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Example. What surfaces are described by these equations?
1. r = 5
2. ✓ =⇡4
3. z = 1
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Example. Use cylindrical coordinates to describe the region above the x-y plane,bounded by the cone z
2 = 4x2 + 4y
2 and the plane z = 6.
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Example. Find the mass of the solid cone bounded by z = 2r and z = 6, if the densityat any point is proportional to its distance from the z-axis.
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Note. Using cylindrical coordinates, if E is a region of space that can be described by:↵ ✓ �, h1(✓) r h2(✓),u1(r,✓) z u2(r,✓), then
ZZZ
E
f (x, y, z) dV =
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
The spherical coordinates of a point P in space are given by (⇢,✓,�), where:
⇢ is ...
✓ is ...
� is ...
For spherical coordinates, there are restrictions on the values of ⇢ and �:
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Example. .
Note. Cartesian coordinates and spherical coordinates are related by:
• x =
• y =
• z =
• ⇢ =
• tan✓ =
• tan� =
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Example. What surfaces are described by these equations?
1. ⇢ = 5
2. ✓ = ⇡4
3. � = ⇡6
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Note. Using spherical coordinates, for a ⇢ b, ↵ ✓ �, � � �,ZZZ
E
f (x, y, z) dV =
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Example. Find the volume of the solid that lies within the sphere x2+ y
2+z2 = 4, above
the x-y plane, and below the cone z =p
x2 + y2.
END OF VIDEO
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Review. The cylindrical coordinates of a point P in space are given by (r,✓, z)
where z is ...
and r and ✓ are ...
As with polar coordinates, r can be positive or negative.
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Review. Cartesian coordinates and cylindrical coordinates are related by:
• x =
• y =
• z =
• r2 =
• tan✓ =
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Example. What surfaces are described by these equations?
1. ✓ = ⇡3
2. r = �2
3. z = r
4. z2 = 4 � r
2
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Example. Find the mass of the solid above the surface z = 4 � x2 � y
2 and inside thesurface x
2 + y2 + z
2 = 16, if the density is given by ⇢(x, y, z) = z2 + 1.
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Example. Rewrite the integral in cylindrical coordinates.
Z 2
0
Z p4�x2
0
Z 6�x2�y
2
px2+y2
xz dz dy dx
Hint: sketch the region, then project it onto the x-y plane and write this in polarcoordinates.
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Review. The spherical coordinates of a point P in space are given by (⇢,✓,�), where:
⇢ is ...
✓ is ...
� is ...
For spherical coordinates, there are restrictions on the values of ⇢ and �:
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Review. Cartesian coordinates and spherical coordinates are related by:
• x =
• y =
• z =
• ⇢ =
• tan✓ =
• tan� =
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Example. What surfaces are described by these equations?
1. � = 2⇡3
2. ⇢ = 3
3. ⇢ = sin(✓) sin(�)
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Note. Using spherical coordinates, for a ⇢ b, ↵ ✓ �, � � �,ZZZ
E
f (x, y, z) dV =
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Example. EvaluateR R R
E
px2 + y2 + z2 dV, where E lies above the cone z =
p3x2 + 3y2
and between the spheres x2 + y
2 + z2 = 1 and x
2 + y2 + z
2 = 4
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Example. Change the integral to spherical coordinates.
Z 2
�2
Z p4�x2
�p
4�x2
Z 2+p
4�x2�y2
2�p
4�x2�y2(x2 + y
2 + z2)3/2
dz dy dx
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§16.5 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES
Extra Example. Find the average distance from a point in a ball of radius a to its center.
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