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Triple IntegralsThis unit is based on Sections 9.15 Chapter 9.All assigned readings and exercises are from the textbookObjectives:Make certain that you can define, and use in context, the terms,

concepts and formulas listed below:• Evaluate triple integrals in Cartesian Coordinates• Express points, surfaces and volumes in cylindrical coordinates• Express points, surfaces and volumes in spherical coordinates• Evaluate triple integrals in cylindrical and spherical coordinates• Evaluate physical characteristics of solids using triple integrals:

volume, center of mass, moment of inertia, total charge, total energy stored in a region…etc.

Reading: Read Section 9.15, pages 539-550.Exercises: Complete problems

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∫∫∫∫∫∫ =DD

dxdydzzyxfdVzyxf ),,(),,(

Triple Integrals in Cartesian CoordinatesThe integral of a function f(x,y,z) over a 3D object D, is given by

The limits on the integration depend on the shape of the body D

dV = dxdydz represents an element of volume

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Triple integrals: limits of integration

∫ ∫ ∫

∫∫∫=

=

=

=

=

= ⎥⎦⎤

⎢⎣⎡ ⎟

⎠⎞⎜

⎝⎛=

bx

ax

xgy

xgy

yxfz

yxfz

D

dxdydzzyxu

dzdydxzyxu

)(

)(

),(

),(

2

1

2

1

),,(

),,(

Assuming we integrate with respect to z, then y, then x, the innermost limits may depend on the other two variables (x and y), the middle limits may depend on the outer variable (z), whereas the outer limits are constants.The main task is to determine the correct limits on x, y, z:

For most engineering applications shapes that are important include: box, cylinder, cone, tetrahedron, sphere.

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Engineering Application of Triple Integrals IVolume V of a region D:

Center of mass for a body D with density ρ(x,y,z)

...~,),,(~

),,(~

==

=

∫ ∫∫

∫ ∫∫

zm

dxdydzzyxyy

m

dxdydzzyxxx

D

D

,

ρ

ρ

∫∫∫=D

dxdydzV

Mass for a body D with density ρ(x,y,z):

∫∫∫=D

dxdydzzyxm ),,(ρ

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Engineering Application of Triple Integrals IIMoment of inertia about the x-axis (Ix) and the y-axis (Iy):

..,),,()(

;),,()(22

22

=+=

+=

∫ ∫∫∫ ∫∫

zDy

Dx

IdxdydzzyxzxI

dxdydzzyxzyI

ρ

ρ

Total charge for a body with charge density ρ(x,y,z)

;),,(∫ ∫∫=D

dxdydzzyxQ ρ

Total electrostatic energy (W) stored in a region with electrostatic filed E

;2

∫ ∫∫=D

dxdydzEkWr +

-

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Example 1 (P9-15-6): Evaluate the integral:

Find the volume bounded by: x=y2, 4- x=y2, z=0 and z=3

Example 3 (P9-15.21)

Example 4 (P9-15.27) Find the center of mass of the solid bounded by: x2+z2=4,, y=0 and y=3 if the density ρ = k y

Example 2 (P9-6.15)

∫ ∫ ∫−4

0

3

0

3/22

0dydxdz

zSketch the region D whose volume V is given by the integral:

Give details of solutions

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Naming convention: a point P(x,y,z) ),,( zr θ⇔

Cylindrical Coordinates

)/(

,

;;sin

cos

22

xy

yxr

zzrrx

1-tan

y ,

=

+=

===

θ

θθ

Relation to Cartesian coordinates (Switching):

r varies from 0 to ∞; θ varies from 0 to 2π, z varies from - ∞ to ∞

Cylindrical coordinates are good for describing solids that are symmetric around an axis.

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sd sd

sd

z

r

addrrtconszadzdrtcons

adzdrtconsr

ndSSdsurfacealDifferenti

ˆ,tanˆ,tan

ˆ,tan

:)ˆ(

θθ

θ

θ

======

=

r

r

r

r

Cylindrical Coordinates

dzddrrdVvolumealDifferenti θ=:

z d

zddrrdr

lengthalDifferenti

ˆˆˆ

:

++= θθlr

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Triple Integrals in Cylindrical Coordinates

θθ

θθθ

βθ

αθ

θ

θ

θ

θdrdrdzzru

dzrdrdzrrfdxdydzzyxf

gr

gr

rfz

rfz

D D

∫ ∫ ∫

∫∫∫ ∫∫∫

=

=

=

=

=

= ⎥⎦⎤

⎢⎣⎡ ⎟

⎠⎞⎜

⎝⎛=

=

)(

)(

),(

),(

1

1

2

1

),,(

),sin,cos(),,(

z-firs

t

θ-last

express dV as

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Example 1 (P9-15-45): Convert the following equation to cylindrical coordinates:

Example 2 (P9-6.51) Use triple integrals in cylindrical coordinates to find the volume V bounded by:

1222 =−+ zyx

0,16,4 22222 ==++=+ zzyxyx

Give details of solutions

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Spherical Coordinates

Naming convention: a point P(x,y,z) ),,( θφρ⇔

)/(tan),/(

,

cossinsincossin

221

222

zyxxy

zyx

x

+==

++=

===

−φθ

ρ

φρθφρθφρ

tan

z ,y ,

1-

Relation to Cartesian coordinates (switching):

ρ varies from 0 to ∞; φ varies from 0 to πθ varies from 0 to 2π,

Spherical coordinates are good for describing solids that are symmetric around the point.

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Spherical Coordinates

sin

,

θφρφρ ddddV

volumealdifferenti2=

d

sin d

sin d

θ

φ

ρ

φρρθ

θρφρφ

θφφρρ

addStcons

addStcons

addStcons

ndSSdsurfacealDifferenti

ˆ,tan

ˆ,tan

ˆ,tan

)ˆ(2

==

==

==

=

r

r

r

r

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Triple Integrals in Spherical Coordinates

φφρρθθφρ

φρθφρθφρ

θφρφρθφρ

ddduI

zyxSubstitue

dddudxdydzzyxfD D

sin),,(

cos,sinsin,cossin:

sin),,(),,(

2

2

∫ ∫ ∫

∫∫∫ ∫∫∫

⎥⎦⎤

⎢⎣⎡ ⎟

⎠⎞⎜

⎝⎛=

===

=

θ-first φ-last

express dV as

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Example 3 (P9-15.82)

Find the moment of inertia about the z axis of the solid:

Example 1 (P9-15-69): Convert the following equation to spherical coordinates:

222 33 yxz +=

2222 azyx =++ The density ρv = kρ.

sCoordinate Spherical ⇒+= ∫ ∫∫ ;),,()( 22

V vz dxdydzzyxyxI ρ

Example 2 (P9-15.76): Find the volume bounded by:

Octant First ,0,3,,4222 ====++ zxyxyzyx

Give details of solutions

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[ ] [ ]θφρ

φφφρ

ρρρ

ρ θφρ ∂

∂+

∂∂

+∂∂

=∇=EEEEv sin

1sinsin11. 2

2

r

Given the electrostatic field ρρ akE ˆ2=r

Calculate the total charge and stored energy in a region bounded by: 0 § ρ § 1, 0 § φ § π and 0 § θ § 2π

[ ] [ ]r

EEr

rErr

E zrv ∂

∂+

∂∂

+∂∂

=∇= θθρ 11.

r

Given the electrostatic field rakrE ˆ3=r

Calculate the total charge and stored energy in a region bounded by: 0 § r § 1, 0 § θ § 2π and 0 § z § 3

Calculate the charge density ρv:

Calculate the charge density ρv:

Optional Homework:

Optional Homework:

Introduce the del operator in both cylindrical and spherical coordinates through these examples.