vector calculus - iith.ac.insuku/vectorcalculus2016/lecture4.pdf · vector calculus dr. d. sukumar...
TRANSCRIPT
![Page 1: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/1.jpg)
Vector Calculus
Dr. D. Sukumar
January 10, 2016
![Page 2: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/2.jpg)
Multiple Integrals
Rectangular Co-ordinates
![Page 3: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/3.jpg)
Multiple Integrals
I Multiple integral = integration of multi-variable functions.
I The process of evaluation leads to multiple evaluation ofsingle variables.
I Integrating the functions
f : R ⊆ R2 → R
Where R is a region in the plane.
I
f : R2 → R
What is integration of constant function?
![Page 4: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/4.jpg)
Multiple Integrals
I Multiple integral = integration of multi-variable functions.
I The process of evaluation leads to multiple evaluation ofsingle variables.
I Integrating the functions
f : R ⊆ R2 → R
Where R is a region in the plane.
I
f : R2 → R
What is integration of constant function?
![Page 5: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/5.jpg)
Multiple Integrals
I Multiple integral = integration of multi-variable functions.
I The process of evaluation leads to multiple evaluation ofsingle variables.
I Integrating the functions
f : R ⊆ R2 → R
Where R is a region in the plane.
I
f : R2 → R
What is integration of constant function?
![Page 6: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/6.jpg)
Multiple Integrals
I Multiple integral = integration of multi-variable functions.
I The process of evaluation leads to multiple evaluation ofsingle variables.
I Integrating the functions
f : R ⊆ R2 → R
Where R is a region in the plane.
I
f : R2 → R
What is integration of constant function?
![Page 7: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/7.jpg)
I Let us consider the function of two variables f (x , y) in theregion R
I Initially we restrict to rectangular regions,
I General regions later.
I First decode what is the domain of the function andco-domain of the function. f : R → R where R is the region.
R := {(x , y) : a ≤ x ≤ b, c ≤ y ≤ d}
![Page 8: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/8.jpg)
I Let us consider the function of two variables f (x , y) in theregion R
I Initially we restrict to rectangular regions,
I General regions later.
I First decode what is the domain of the function andco-domain of the function. f : R → R where R is the region.
R := {(x , y) : a ≤ x ≤ b, c ≤ y ≤ d}
![Page 9: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/9.jpg)
I Let us consider the function of two variables f (x , y) in theregion R
I Initially we restrict to rectangular regions,
I General regions later.
I First decode what is the domain of the function andco-domain of the function. f : R → R where R is the region.
R := {(x , y) : a ≤ x ≤ b, c ≤ y ≤ d}
![Page 10: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/10.jpg)
I Let us consider the function of two variables f (x , y) in theregion R
I Initially we restrict to rectangular regions,
I General regions later.
I First decode what is the domain of the function andco-domain of the function. f : R → R where R is the region.
R := {(x , y) : a ≤ x ≤ b, c ≤ y ≤ d}
![Page 11: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/11.jpg)
convergence
I Since intuitively, plane (domain) has two dimension and(co-domain) has one dimension, we can realize the function inthree dimension.
I The base is in two dimension (so small rectangles) and theheight as the function value.
I Let the small area be ∆A0 and height f (x0, y0) so theexpression f (x0, y0)A0 represents the volume.
I Do this process in the whole region and we will end up withintegration of the function f (x , y).
Sn =n∑
k=1
f (xk , yk)∆AK
![Page 12: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/12.jpg)
convergence
I Since intuitively, plane (domain) has two dimension and(co-domain) has one dimension, we can realize the function inthree dimension.
I The base is in two dimension (so small rectangles) and theheight as the function value.
I Let the small area be ∆A0 and height f (x0, y0) so theexpression f (x0, y0)A0 represents the volume.
I Do this process in the whole region and we will end up withintegration of the function f (x , y).
Sn =n∑
k=1
f (xk , yk)∆AK
![Page 13: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/13.jpg)
convergence
I Since intuitively, plane (domain) has two dimension and(co-domain) has one dimension, we can realize the function inthree dimension.
I The base is in two dimension (so small rectangles) and theheight as the function value.
I Let the small area be ∆A0 and height f (x0, y0) so theexpression f (x0, y0)A0 represents the volume.
I Do this process in the whole region and we will end up withintegration of the function f (x , y).
Sn =n∑
k=1
f (xk , yk)∆AK
![Page 14: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/14.jpg)
convergence
I Since intuitively, plane (domain) has two dimension and(co-domain) has one dimension, we can realize the function inthree dimension.
I The base is in two dimension (so small rectangles) and theheight as the function value.
I Let the small area be ∆A0 and height f (x0, y0) so theexpression f (x0, y0)A0 represents the volume.
I Do this process in the whole region and we will end up withintegration of the function f (x , y).
Sn =n∑
k=1
f (xk , yk)∆AK
![Page 15: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/15.jpg)
Convergence
If Sn converges to some value after refining the mesh (base area)width and depth, then the value is called the double integral of fover R and denoted by
¨Rf (x , y)dA = lim
n→∞
n∑k=1
f (xk , yk)∆AK
I When the refinement really converges? - integration theory.
I But we assume something more ’continuity’ of the function.Though the property is not necessary. This integrals have lotof properties like the single integrals.
![Page 16: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/16.jpg)
Convergence
If Sn converges to some value after refining the mesh (base area)width and depth, then the value is called the double integral of fover R and denoted by
¨Rf (x , y)dA = lim
n→∞
n∑k=1
f (xk , yk)∆AK
I When the refinement really converges? - integration theory.
I But we assume something more ’continuity’ of the function.Though the property is not necessary. This integrals have lotof properties like the single integrals.
![Page 17: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/17.jpg)
Convergence
If Sn converges to some value after refining the mesh (base area)width and depth, then the value is called the double integral of fover R and denoted by
¨Rf (x , y)dA = lim
n→∞
n∑k=1
f (xk , yk)∆AK
I When the refinement really converges? - integration theory.
I But we assume something more ’continuity’ of the function.Though the property is not necessary. This integrals have lotof properties like the single integrals.
![Page 18: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/18.jpg)
Convergence
If Sn converges to some value after refining the mesh (base area)width and depth, then the value is called the double integral of fover R and denoted by
¨Rf (x , y)dA = lim
n→∞
n∑k=1
f (xk , yk)∆AK
I When the refinement really converges? - integration theory.
I But we assume something more ’continuity’ of the function.Though the property is not necessary. This integrals have lotof properties like the single integrals.
![Page 19: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/19.jpg)
Method
We can understand or visualize the double integrals as volumes
∆Ak Small rectangular region (in X Y plane)
f (xk , yk) Height of rectangular prism with ∆Ak as the base (inZ plane)
V Volume under the surface f (x , y)
V =
¨Rf (x , y)dA = lim
n→∞
n∑k=1
f (xk , yk)∆AK
![Page 20: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/20.jpg)
Method
We can understand or visualize the double integrals as volumes
∆Ak Small rectangular region (in X Y plane)
f (xk , yk) Height of rectangular prism with ∆Ak as the base (inZ plane)
V Volume under the surface f (x , y)
V =
¨Rf (x , y)dA = lim
n→∞
n∑k=1
f (xk , yk)∆AK
![Page 21: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/21.jpg)
Method
We can understand or visualize the double integrals as volumes
∆Ak Small rectangular region (in X Y plane)
f (xk , yk) Height of rectangular prism with ∆Ak as the base (inZ plane)
V Volume under the surface f (x , y)
V =
¨Rf (x , y)dA = lim
n→∞
n∑k=1
f (xk , yk)∆AK
![Page 22: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/22.jpg)
Method
We can understand or visualize the double integrals as volumes
∆Ak Small rectangular region (in X Y plane)
f (xk , yk) Height of rectangular prism with ∆Ak as the base (inZ plane)
V Volume under the surface f (x , y)
V =
¨Rf (x , y)dA = lim
n→∞
n∑k=1
f (xk , yk)∆AK
![Page 23: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/23.jpg)
Method
We can understand or visualize the double integrals as volumes
∆Ak Small rectangular region (in X Y plane)
f (xk , yk) Height of rectangular prism with ∆Ak as the base (inZ plane)
V Volume under the surface f (x , y)
V =
¨Rf (x , y)dA = lim
n→∞
n∑k=1
f (xk , yk)∆AK
![Page 24: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/24.jpg)
Method
This volume coincides with the volume that we find by slicing thevolume into areas.
The idea
Double integral = Two single integrals
The equality is given by Fubuni’s Theorem. Let us understand thiswith a simple example.
Example
Calculate the volume under the plane z = f (x , y) = 4 + x − y overthe rectangular region
R = 0 ≤ x ≤ 1, 0 ≤ y ≤ 2
Let us first try to see the surface.
![Page 25: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/25.jpg)
Method
This volume coincides with the volume that we find by slicing thevolume into areas.The idea
Double integral = Two single integrals
The equality is given by Fubuni’s Theorem. Let us understand thiswith a simple example.
Example
Calculate the volume under the plane z = f (x , y) = 4 + x − y overthe rectangular region
R = 0 ≤ x ≤ 1, 0 ≤ y ≤ 2
Let us first try to see the surface.
![Page 26: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/26.jpg)
Method
This volume coincides with the volume that we find by slicing thevolume into areas.The idea
Double integral = Two single integrals
The equality is given by Fubuni’s Theorem. Let us understand thiswith a simple example.
Example
Calculate the volume under the plane z = f (x , y) = 4 + x − y overthe rectangular region
R = 0 ≤ x ≤ 1, 0 ≤ y ≤ 2
Let us first try to see the surface.
![Page 27: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/27.jpg)
Method
This volume coincides with the volume that we find by slicing thevolume into areas.The idea
Double integral = Two single integrals
The equality is given by Fubuni’s Theorem. Let us understand thiswith a simple example.
Example
Calculate the volume under the plane z = f (x , y) = 4 + x − y overthe rectangular region
R = 0 ≤ x ≤ 1, 0 ≤ y ≤ 2
Let us first try to see the surface.
![Page 28: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/28.jpg)
Problem
Let us do the slicing in x-direction. For a fixed x we will have
To get the volume, we should integrate over such all possibleareas.
´ 10 A(x)dx Look out only A(x) and see that
A(x) =
ˆ 2
0(4 + x − y)dy
Hence the volume under the plane is given by
ˆ 1
0
(ˆ 2
0(4 + x − y)dy
)dx =
ˆ 1
0
[4y + xy − y2
2
]20
dx
=
ˆ 1
0(6 + 2x)dx
=[6x + x2
]10
= 7Units
![Page 29: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/29.jpg)
Problem
Let us do the slicing in x-direction. For a fixed x we will haveTo get the volume, we should integrate over such all possibleareas.
´ 10 A(x)dx Look out only A(x) and see that
A(x) =
ˆ 2
0(4 + x − y)dy
Hence the volume under the plane is given by
ˆ 1
0
(ˆ 2
0(4 + x − y)dy
)dx =
ˆ 1
0
[4y + xy − y2
2
]20
dx
=
ˆ 1
0(6 + 2x)dx
=[6x + x2
]10
= 7Units
![Page 30: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/30.jpg)
Problem
Let us do the slicing in x-direction. For a fixed x we will haveTo get the volume, we should integrate over such all possibleareas.
´ 10 A(x)dx Look out only A(x) and see that
A(x) =
ˆ 2
0(4 + x − y)dy
Hence the volume under the plane is given by
ˆ 1
0
(ˆ 2
0(4 + x − y)dy
)dx =
ˆ 1
0
[4y + xy − y2
2
]20
dx
=
ˆ 1
0(6 + 2x)dx
=[6x + x2
]10
= 7Units
![Page 31: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/31.jpg)
Slicing in other direction
Doing it in the other direction, that is in y - direction. For a fixed ywe will have the picture.
To get the volume, we should integrateover such all possible areas.
´ 20 A(y)dy Look out only A(y) and
see that
A(y) =
ˆ 1
0(4 + x − y)dx
Hence the volume under the plane is given by
ˆ 2
0
(ˆ 1
0(4 + x − y)dx
)dy =
ˆ 2
0
[4x +
x2
2− xy
]10
dx
=
ˆ 2
0
(9
2− y
)dy
=
[9
2y − y2
2
]20
= 7Units
![Page 32: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/32.jpg)
Slicing in other direction
Doing it in the other direction, that is in y - direction. For a fixed ywe will have the picture. To get the volume, we should integrateover such all possible areas.
´ 20 A(y)dy Look out only A(y) and
see that
A(y) =
ˆ 1
0(4 + x − y)dx
Hence the volume under the plane is given by
ˆ 2
0
(ˆ 1
0(4 + x − y)dx
)dy =
ˆ 2
0
[4x +
x2
2− xy
]10
dx
=
ˆ 2
0
(9
2− y
)dy
=
[9
2y − y2
2
]20
= 7Units
![Page 33: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/33.jpg)
Rectangle Co-ordinates
Figure: x =constant, y =constant
![Page 34: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/34.jpg)
Rectangle Co-ordinates
Figure: x =constant,
y =constant
![Page 35: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/35.jpg)
Rectangle Co-ordinates
Figure: x =constant, y =constant
![Page 36: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/36.jpg)
Fubini’s Theorem(first form)(1907)
If f (x , y) be a continuous function on a rectangular regionR := {(x , y) : a ≤ x ≤ b, c ≤ y ≤ d} then
¨Rf (x , y)dA =
ˆ b
a
ˆ d
cf (x , y)dydx =
ˆ d
c
ˆ b
af (x , y)dxdy
![Page 37: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/37.jpg)
Fubini’s Theorem(first form)(1907)
If f (x , y) be a continuous function on a rectangular regionR := {(x , y) : a ≤ x ≤ b, c ≤ y ≤ d} then
¨Rf (x , y)dA
=
ˆ b
a
ˆ d
cf (x , y)dydx =
ˆ d
c
ˆ b
af (x , y)dxdy
![Page 38: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/38.jpg)
Fubini’s Theorem(first form)(1907)
If f (x , y) be a continuous function on a rectangular regionR := {(x , y) : a ≤ x ≤ b, c ≤ y ≤ d} then
¨Rf (x , y)dA =
ˆ b
a
ˆ d
cf (x , y)dydx
=
ˆ d
c
ˆ b
af (x , y)dxdy
![Page 39: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/39.jpg)
Fubini’s Theorem(first form)(1907)
If f (x , y) be a continuous function on a rectangular regionR := {(x , y) : a ≤ x ≤ b, c ≤ y ≤ d} then
¨Rf (x , y)dA =
ˆ b
a
ˆ d
cf (x , y)dydx =
ˆ d
c
ˆ b
af (x , y)dxdy
![Page 40: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/40.jpg)
Fubini’s Theorem(Stronger form)
If f (x , y) be a continuous function on a rectangular regionR := {(x , y) : a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)} then
¨Rf (x , y)dA =
ˆ b
a
ˆ g2(x)
g1(x)f (x , y)dydx
If f (x , y) be a continuous function on a rectangular regionR := {(x , y) : h1(x) ≤ y ≤ h2(x), c ≤ y ≤ d} then
¨Rf (x , y)dA =
ˆ d
c
ˆ h2(y)
h1(y)f (x , y)dxdy
![Page 41: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/41.jpg)
Fubini’s Theorem(Stronger form)
If f (x , y) be a continuous function on a rectangular regionR := {(x , y) : a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)} then
¨Rf (x , y)dA
=
ˆ b
a
ˆ g2(x)
g1(x)f (x , y)dydx
If f (x , y) be a continuous function on a rectangular regionR := {(x , y) : h1(x) ≤ y ≤ h2(x), c ≤ y ≤ d} then
¨Rf (x , y)dA =
ˆ d
c
ˆ h2(y)
h1(y)f (x , y)dxdy
![Page 42: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/42.jpg)
Fubini’s Theorem(Stronger form)
If f (x , y) be a continuous function on a rectangular regionR := {(x , y) : a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)} then
¨Rf (x , y)dA =
ˆ b
a
ˆ g2(x)
g1(x)f (x , y)dydx
If f (x , y) be a continuous function on a rectangular regionR := {(x , y) : h1(x) ≤ y ≤ h2(x), c ≤ y ≤ d} then
¨Rf (x , y)dA =
ˆ d
c
ˆ h2(y)
h1(y)f (x , y)dxdy
![Page 43: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/43.jpg)
Fubini’s Theorem(Stronger form)
If f (x , y) be a continuous function on a rectangular regionR := {(x , y) : a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)} then
¨Rf (x , y)dA =
ˆ b
a
ˆ g2(x)
g1(x)f (x , y)dydx
If f (x , y) be a continuous function on a rectangular regionR := {(x , y) : h1(x) ≤ y ≤ h2(x), c ≤ y ≤ d} then
¨Rf (x , y)dA
=
ˆ d
c
ˆ h2(y)
h1(y)f (x , y)dxdy
![Page 44: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/44.jpg)
Fubini’s Theorem(Stronger form)
If f (x , y) be a continuous function on a rectangular regionR := {(x , y) : a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)} then
¨Rf (x , y)dA =
ˆ b
a
ˆ g2(x)
g1(x)f (x , y)dydx
If f (x , y) be a continuous function on a rectangular regionR := {(x , y) : h1(x) ≤ y ≤ h2(x), c ≤ y ≤ d} then
¨Rf (x , y)dA =
ˆ d
c
ˆ h2(y)
h1(y)f (x , y)dxdy
![Page 45: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/45.jpg)
Sketch the region of integration and evaluate
1.´ π0
´ sin(x)0 ydydx π/4
tIntegrating functionf (x , y) = y
Region R:0 ≤ y ≤ sin(x)
0 ≤ x ≤ π
2.´ 0−1´ 1−1(x + y + 1)dxdy 1
3.´ 21
´ y2
y dxdy 5/6
![Page 46: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/46.jpg)
Sketch the region of integration and evaluate
1.´ π0
´ sin(x)0 ydydx π/4
tIntegrating functionf (x , y) = y
Region R:0 ≤ y ≤ sin(x)
0 ≤ x ≤ π
2.´ 0−1´ 1−1(x + y + 1)dxdy 1
3.´ 21
´ y2
y dxdy 5/6
![Page 47: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/47.jpg)
Sketch the region of integration and evaluate
1.´ π0
´ sin(x)0 ydydx π/4
tIntegrating functionf (x , y) = y
Region R:0 ≤ y ≤ sin(x)
0 ≤ x ≤ π
2.´ 0−1´ 1−1(x + y + 1)dxdy 1
3.´ 21
´ y2
y dxdy 5/6
![Page 48: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/48.jpg)
Sketch the region of integration and write an equivalent doubleintegral with the order of integration reversed
1.´ 20
´ 0y−2 dxdy
2.´ 20
´ 4−y2
0 ydxdy
3.´ 20
´ √4−x2−√4−x2 dxdy
![Page 49: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/49.jpg)
1. Find the volume of the region that lies under the paraboloidz = x2 + y2 and above the triangle enclosed by the linesy = x , x = 0 and x + y = 2 in the XY -plane. 4/3
2. What region R in the XY -plane minimizes the value of
ˆ ˆ(x2 + y2 − 9)dA
![Page 50: Vector Calculus - iith.ac.insuku/vectorcalculus2016/Lecture4.pdf · Vector Calculus Dr. D. Sukumar January 10, 2016. Multiple Integrals Rectangular Co-ordinates. Multiple Integrals](https://reader031.vdocuments.site/reader031/viewer/2022022605/5b78d4fd7f8b9a7f378c4fb9/html5/thumbnails/50.jpg)
Sketch the regions bounded by the given lines and curves. Thenexpress the regions area as an iterated double integrals andevaluate the integral
1. Co-ordinate axes and the line x + y = 2 2
2. The curve y = ex and the lines y = 0, x = 0 and x = ln 2 1
3. The parabolas x = y2 − 1 and x = 2y2 − 2 4/3
4.´ 30
´ x(2−x)−x dydx