triple integrals and 3d coordinates 9-15users.encs.concordia.ca/~rbhat/engr233/triple integrals and...
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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.
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