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Triple Integrals — §12.5 110 Triple Integrals For a function f (x , y , z ) defined over a bounded region E in three dimensions, we can take the triple integral ZZZ E f (x , y , z ) dV

Triple Integrals | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

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Page 1: Triple Integrals | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 110

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 110

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 110

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 110

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 110

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 110

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 110

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 111

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 111

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 111

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 112

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 112

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 112

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 112

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 112

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 113

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 113

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 113

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 113

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 114

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 114

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 114

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 114

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 114

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 114

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 114

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 114

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 114

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 114

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 115

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 115

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 115

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 115

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 | 12.5 110 Triple Integralsqcpages.qc.cuny.edu/~chanusa/courses/201/14f/notes/201fa14ch125c.pdf Triple Integrals | x12.5 110 Triple Integrals For a function f(x;y;z)

Triple Integrals — §12.5 116

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 .