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![Page 1: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/1.jpg)
Triple Integrals — §12.5 111
Triple Integrals
For a function f (x , y , z) defined over a bounded region E in threedimensions, we can take the triple integral∫∫∫
Ef (x , y , z) dV
If f is continuous over a region that is a box
B = [a, b]× [c, d ]× [r , s],
Fubini’s theorem says that
![Page 2: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/2.jpg)
Triple Integrals — §12.5 111
Triple Integrals
For a function f (x , y , z) defined over a bounded region E in threedimensions, we can take the triple integral∫∫∫
Ef (x , y , z) dV
If f is continuous over a region that is a box
B = [a, b]× [c, d ]× [r , s],
Fubini’s theorem says that∫∫∫Bf (x , y , z) dV =
∫ z=s
z=r
∫ y=d
y=c
∫ x=b
x=af (x , y , z) dx dy dz
![Page 3: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/3.jpg)
Triple Integrals — §12.5 111
Triple Integrals
For a function f (x , y , z) defined over a bounded region E in threedimensions, we can take the triple integral∫∫∫
Ef (x , y , z) dV
If f is continuous over a region that is a box
B = [a, b]× [c, d ]× [r , s],
Fubini’s theorem says that∫∫∫Bf (x , y , z) dV =
∫ z=s
z=r
∫ y=d
y=c
∫ x=b
x=af (x , y , z) dx dy dz
![Page 4: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/4.jpg)
Triple Integrals — §12.5 111
Triple Integrals
For a function f (x , y , z) defined over a bounded region E in threedimensions, we can take the triple integral∫∫∫
Ef (x , y , z) dV
If f is continuous over a region that is a box
B = [a, b]× [c, d ]× [r , s],
Fubini’s theorem says that∫∫∫Bf (x , y , z) dV =
∫ z=s
z=r
∫ y=d
y=c
∫ x=b
x=af (x , y , z) dx dy dz
And that you are allowed to choose the order of integration you wish.
![Page 5: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/5.jpg)
Triple Integrals — §12.5 111
Triple Integrals
For a function f (x , y , z) defined over a bounded region E in threedimensions, we can take the triple integral∫∫∫
Ef (x , y , z) dV
If f is continuous over a region that is a box
B = [a, b]× [c, d ]× [r , s],
Fubini’s theorem says that∫∫∫Bf (x , y , z) dV =
∫ y=d
y=c
∫ x=b
x=a
∫ z=s
z=rf (x , y , z) dz dx dy
And that you are allowed to choose the order of integration you wish.
![Page 6: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/6.jpg)
Triple Integrals — §12.5 111
Triple Integrals
For a function f (x , y , z) defined over a bounded region E in threedimensions, we can take the triple integral∫∫∫
Ef (x , y , z) dV
If f is continuous over a region that is a box
B = [a, b]× [c, d ]× [r , s],
Fubini’s theorem says that∫∫∫Bf (x , y , z) dV =
∫ y=d
y=c
∫ z=s
z=r
∫ x=b
x=af (x , y , z) dx dz dy
And that you are allowed to choose the order of integration you wish.
![Page 7: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/7.jpg)
Triple Integrals — §12.5 111
Triple Integrals
For a function f (x , y , z) defined over a bounded region E in threedimensions, we can take the triple integral∫∫∫
Ef (x , y , z) dV
If f is continuous over a region that is a box
B = [a, b]× [c, d ]× [r , s],
Fubini’s theorem says that∫∫∫Bf (x , y , z) dV =
∫ x=b
x=a
∫ y=d
y=c
∫ z=s
z=rf (x , y , z) dz dy dx
And that you are allowed to choose the order of integration you wish.
![Page 8: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/8.jpg)
Triple Integrals — §12.5 112
Example
Example. Find∫∫∫
B xyz2 dV for B = [0, 1]× [−1, 2]× [0, 3]
Solution.
The difficulties arise when
I Regions are not boxes. (Today)
I Regions are best defined in polar-like coordinates. (Next time)
![Page 9: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/9.jpg)
Triple Integrals — §12.5 112
Example
Example. Find∫∫∫
B xyz2 dV for B = [0, 1]× [−1, 2]× [0, 3]
Solution.
The difficulties arise when
I Regions are not boxes. (Today)
I Regions are best defined in polar-like coordinates. (Next time)
![Page 10: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/10.jpg)
Triple Integrals — §12.5 112
Example
Example. Find∫∫∫
B xyz2 dV for B = [0, 1]× [−1, 2]× [0, 3]
Solution.
The difficulties arise when
I Regions are not boxes. (Today)
I Regions are best defined in polar-like coordinates. (Next time)
![Page 11: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/11.jpg)
Triple Integrals — §12.5 113
Setting up complicated triple integrals
I We only consider triple integrals over regions that can bedefined as being between two surfaces.
I This allows us to reduce our triple integral to a double integral.(Which may itself be complicated...)
Three types:∫∫∫Ef dV =
∫∫D
[∫ z=u2(x ,y)
z=u1(x ,y)f (x , y , z) dz
]dA
Typ
e1
∫∫∫Ef dV =
∫∫D
[∫ x=u2(y ,z)
x=u1(y ,z)f (x , y , z) dx
]dA
Typ
e2
∫∫∫Ef dV =
∫∫D
[∫ y=u2(x ,z)
y=u1(x ,z)f (x , y , z) dy
]dA
Typ
e3
![Page 12: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/12.jpg)
Triple Integrals — §12.5 113
Setting up complicated triple integrals
I We only consider triple integrals over regions that can bedefined as being between two surfaces.
I This allows us to reduce our triple integral to a double integral.(Which may itself be complicated...)
Three types:∫∫∫Ef dV =
∫∫D
[∫ z=u2(x ,y)
z=u1(x ,y)f (x , y , z) dz
]dA
Typ
e1
∫∫∫Ef dV =
∫∫D
[∫ x=u2(y ,z)
x=u1(y ,z)f (x , y , z) dx
]dA
Typ
e2
∫∫∫Ef dV =
∫∫D
[∫ y=u2(x ,z)
y=u1(x ,z)f (x , y , z) dy
]dA
Typ
e3
![Page 13: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/13.jpg)
Triple Integrals — §12.5 113
Setting up complicated triple integrals
I We only consider triple integrals over regions that can bedefined as being between two surfaces.
I This allows us to reduce our triple integral to a double integral.(Which may itself be complicated...)
Three types:∫∫∫Ef dV =
∫∫D
[∫ z=u2(x ,y)
z=u1(x ,y)f (x , y , z) dz
]dA
Typ
e1
∫∫∫Ef dV =
∫∫D
[∫ x=u2(y ,z)
x=u1(y ,z)f (x , y , z) dx
]dA
Typ
e2
∫∫∫Ef dV =
∫∫D
[∫ y=u2(x ,z)
y=u1(x ,z)f (x , y , z) dy
]dA
Typ
e3
![Page 14: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/14.jpg)
Triple Integrals — §12.5 113
Setting up complicated triple integrals
I We only consider triple integrals over regions that can bedefined as being between two surfaces.
I This allows us to reduce our triple integral to a double integral.(Which may itself be complicated...)
Three types:∫∫∫Ef dV =
∫∫D
[∫ z=u2(x ,y)
z=u1(x ,y)f (x , y , z) dz
]dA
Typ
e1
∫∫∫Ef dV =
∫∫D
[∫ x=u2(y ,z)
x=u1(y ,z)f (x , y , z) dx
]dA
Typ
e2
∫∫∫Ef dV =
∫∫D
[∫ y=u2(x ,z)
y=u1(x ,z)f (x , y , z) dy
]dA
Typ
e3
![Page 15: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/15.jpg)
Triple Integrals — §12.5 113
Setting up complicated triple integrals
I We only consider triple integrals over regions that can bedefined as being between two surfaces.
I This allows us to reduce our triple integral to a double integral.(Which may itself be complicated...)
Three types:∫∫∫Ef dV =
∫∫D
[∫ z=u2(x ,y)
z=u1(x ,y)f (x , y , z) dz
]dA
Typ
e1
∫∫∫Ef dV =
∫∫D
[∫ x=u2(y ,z)
x=u1(y ,z)f (x , y , z) dx
]dA
Typ
e2
∫∫∫Ef dV =
∫∫D
[∫ y=u2(x ,z)
y=u1(x ,z)f (x , y , z) dy
]dA
Typ
e3
![Page 16: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/16.jpg)
Triple Integrals — §12.5 114
Triple Integral Strategies
The hard part is figuring out the bounds of your integrals.
I Project your region E onto one of the xy -, yz-, or xz-planes, anduse the boundary of this projection to find bounds on domain D.
I Over this domain D, the region E is defined by some“higher function” and some “lower function”.These give the bounds on the innermost integral.
You may need to try multiple projections to find the easiest integralto integrate. Then use all the tools in your toolbox to integrate it.
![Page 17: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/17.jpg)
Triple Integrals — §12.5 114
Triple Integral Strategies
The hard part is figuring out the bounds of your integrals.
I Project your region E onto one of the xy -, yz-, or xz-planes, anduse the boundary of this projection to find bounds on domain D.
I Over this domain D, the region E is defined by some“higher function” and some “lower function”.These give the bounds on the innermost integral.
You may need to try multiple projections to find the easiest integralto integrate. Then use all the tools in your toolbox to integrate it.
![Page 18: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/18.jpg)
Triple Integrals — §12.5 114
Triple Integral Strategies
The hard part is figuring out the bounds of your integrals.
I Project your region E onto one of the xy -, yz-, or xz-planes, anduse the boundary of this projection to find bounds on domain D.
I Over this domain D, the region E is defined by some“higher function” and some “lower function”.These give the bounds on the innermost integral.
You may need to try multiple projections to find the easiest integralto integrate. Then use all the tools in your toolbox to integrate it.
![Page 19: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/19.jpg)
Triple Integrals — §12.5 114
Triple Integral Strategies
The hard part is figuring out the bounds of your integrals.
I Project your region E onto one of the xy -, yz-, or xz-planes, anduse the boundary of this projection to find bounds on domain D.
I Over this domain D, the region E is defined by some“higher function” and some “lower function”.These give the bounds on the innermost integral.
You may need to try multiple projections to find the easiest integralto integrate. Then use all the tools in your toolbox to integrate it.
![Page 20: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/20.jpg)
Triple Integrals — §12.5 115
Example
Example. Evaluate∫∫∫
E
√x2 + z2 dV where E is the region
bounded by the hyperboloid y = x2 + z2 and the plane y = 4.
Project onto xy -plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =
∫ 2
−2
∫ 4
x2
(∫ √y−x2
−√
y−x2
√x2 + z2 dz
)
dy dx
Project onto xz-plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =
∫∫D=circle
x2+z2=4
(∫ 4
x2+z2
√x2 + z2 dy
)
dA
![Page 21: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/21.jpg)
Triple Integrals — §12.5 115
Example
Example. Evaluate∫∫∫
E
√x2 + z2 dV where E is the region
bounded by the hyperboloid y = x2 + z2 and the plane y = 4.
Project onto xy -plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =
∫ 2
−2
∫ 4
x2
(∫ √y−x2
−√
y−x2
√x2 + z2 dz
)
dy dx
Project onto xz-plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =
∫∫D=circle
x2+z2=4
(∫ 4
x2+z2
√x2 + z2 dy
)
dA
![Page 22: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/22.jpg)
Triple Integrals — §12.5 115
Example
Example. Evaluate∫∫∫
E
√x2 + z2 dV where E is the region
bounded by the hyperboloid y = x2 + z2 and the plane y = 4.
Project onto xy -plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =
∫ 2
−2
∫ 4
x2
(∫ √y−x2
−√
y−x2
√x2 + z2 dz
)
dy dx
Project onto xz-plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =
∫∫D=circle
x2+z2=4
(∫ 4
x2+z2
√x2 + z2 dy
)
dA
![Page 23: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/23.jpg)
Triple Integrals — §12.5 115
Example
Example. Evaluate∫∫∫
E
√x2 + z2 dV where E is the region
bounded by the hyperboloid y = x2 + z2 and the plane y = 4.
Project onto xy -plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =
∫ 2
−2
∫ 4
x2
(∫ √y−x2
−√
y−x2
√x2 + z2 dz
)
dy dx
Project onto xz-plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =
∫∫D=circle
x2+z2=4
(∫ 4
x2+z2
√x2 + z2 dy
)
dA
![Page 24: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/24.jpg)
Triple Integrals — §12.5 115
Example
Example. Evaluate∫∫∫
E
√x2 + z2 dV where E is the region
bounded by the hyperboloid y = x2 + z2 and the plane y = 4.
Project onto xy -plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =
∫ 2
−2
∫ 4
x2
(∫ √y−x2
−√
y−x2
√x2 + z2 dz
)
dy dx
Project onto xz-plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =
∫∫D=circle
x2+z2=4
(∫ 4
x2+z2
√x2 + z2 dy
)
dA
![Page 25: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/25.jpg)
Triple Integrals — §12.5 115
Example
Example. Evaluate∫∫∫
E
√x2 + z2 dV where E is the region
bounded by the hyperboloid y = x2 + z2 and the plane y = 4.
Project onto xy -plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =∫ 2
−2
∫ 4
x2
(∫ √y−x2
−√
y−x2
√x2 + z2 dz
)dy dx
Project onto xz-plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =
∫∫D=circle
x2+z2=4
(∫ 4
x2+z2
√x2 + z2 dy
)
dA
![Page 26: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/26.jpg)
Triple Integrals — §12.5 115
Example
Example. Evaluate∫∫∫
E
√x2 + z2 dV where E is the region
bounded by the hyperboloid y = x2 + z2 and the plane y = 4.
Project onto xy -plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =∫ 2
−2
∫ 4
x2
(∫ √y−x2
−√
y−x2
√x2 + z2 dz
)dy dx
Project onto xz-plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =
∫∫D=circle
x2+z2=4
(∫ 4
x2+z2
√x2 + z2 dy
)
dA
![Page 27: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/27.jpg)
Triple Integrals — §12.5 115
Example
Example. Evaluate∫∫∫
E
√x2 + z2 dV where E is the region
bounded by the hyperboloid y = x2 + z2 and the plane y = 4.
Project onto xy -plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =∫ 2
−2
∫ 4
x2
(∫ √y−x2
−√
y−x2
√x2 + z2 dz
)dy dx
Project onto xz-plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =
∫∫D=circle
x2+z2=4
(∫ 4
x2+z2
√x2 + z2 dy
)
dA
![Page 28: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/28.jpg)
Triple Integrals — §12.5 115
Example
Example. Evaluate∫∫∫
E
√x2 + z2 dV where E is the region
bounded by the hyperboloid y = x2 + z2 and the plane y = 4.
Project onto xy -plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =∫ 2
−2
∫ 4
x2
(∫ √y−x2
−√
y−x2
√x2 + z2 dz
)dy dx
Project onto xz-plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =
∫∫D=circle
x2+z2=4
(∫ 4
x2+z2
√x2 + z2 dy
)
dA
![Page 29: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/29.jpg)
Triple Integrals — §12.5 115
Example
Example. Evaluate∫∫∫
E
√x2 + z2 dV where E is the region
bounded by the hyperboloid y = x2 + z2 and the plane y = 4.
Project onto xy -plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =∫ 2
−2
∫ 4
x2
(∫ √y−x2
−√
y−x2
√x2 + z2 dz
)dy dx
Project onto xz-plane
D is defined by
The higher and lower functions are
∫∫∫E
√x2 + z2 dV =∫∫
D=circle
x2+z2=4
(∫ 4
x2+z2
√x2 + z2 dy
)dA
![Page 30: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/30.jpg)
Triple Integrals — §12.5 116
Example, continued
Now calculate
∫∫D=circle
x2+z2=4
(∫ 4
x2+z2
√x2 + z2 dy
)dA
There is no y , so the innermost integral is easy:
=
∫∫D=circle
x2+z2=4
(√x2 + z2 · y
]4x2+z2 dA
=
∫∫D=circle
x2+z2=4
√x2 + z2 ·
(4− x2 − z2
)dA
This integral is easier to do using
![Page 31: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/31.jpg)
Triple Integrals — §12.5 116
Example, continued
Now calculate
∫∫D=circle
x2+z2=4
(∫ 4
x2+z2
√x2 + z2 dy
)dA
There is no y , so the innermost integral is easy:
=
∫∫D=circle
x2+z2=4
(√x2 + z2 · y
]4x2+z2 dA
=
∫∫D=circle
x2+z2=4
√x2 + z2 ·
(4− x2 − z2
)dA
This integral is easier to do using
![Page 32: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/32.jpg)
Triple Integrals — §12.5 116
Example, continued
Now calculate
∫∫D=circle
x2+z2=4
(∫ 4
x2+z2
√x2 + z2 dy
)dA
There is no y , so the innermost integral is easy:
=
∫∫D=circle
x2+z2=4
(√x2 + z2 · y
]4x2+z2 dA
=
∫∫D=circle
x2+z2=4
√x2 + z2 ·
(4− x2 − z2
)dA
This integral is easier to do using
![Page 33: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/33.jpg)
Triple Integrals — §12.5 116
Example, continued
Now calculate
∫∫D=circle
x2+z2=4
(∫ 4
x2+z2
√x2 + z2 dy
)dA
There is no y , so the innermost integral is easy:
=
∫∫D=circle
x2+z2=4
(√x2 + z2 · y
]4x2+z2 dA
=
∫∫D=circle
x2+z2=4
√x2 + z2 ·
(4− x2 − z2
)dA
This integral is easier to do using
![Page 34: Triple Integrals | 12.5 111 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/17/notes/201fa17ch125c.pdfTriple Integrals | x12.5 111 Triple Integrals For a function f(x;y;z)](https://reader035.fdocuments.in/reader035/viewer/2022070616/5d160a9488c993a82b8bd52e/html5/thumbnails/34.jpg)
Triple Integrals — §12.5 117
Loose ends
Density in three dimensions
I Given a mass density function ρ(x , y , z) (mass per unit volume)
mass =
∫∫∫Eρ(x , y , z) dV .
Average value in three dimensions
I The average value of a function f (x , y , z) over a region E is
fave =1
V (E )
∫∫∫Ef (x , y , z) dV .