Triple Integrals in Cylindrical& Spherical Coordinates
Math 55 - Elementary Analysis III
Institute of MathematicsUniversity of the Philippines
Diliman
Math 55 Cylindrical & Spherical Coordinates 1/ 23
Cylindrical Coordinates
Consider a point P in space. Let Q be the projection of P ontothe xy-plane. Let r be the distance of Q from the origin, and be the angle between the positive x-axis and the line segmentOQ.
Qr
x
y
z
P
O
The ordered triple (r, , z) is called the cylindrical coordinatesof P .
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Surfaces in Cylindrical Coordinates
If r = c, a constant then we have a cylinder with radius |c|.
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Surfaces in Cylindrical Coordinates
If = c, a constant then we have a half-plane.
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Surfaces in Cylindrical Coordinates
If z = r, then we have a cone.
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Cartesian Cylindrical
r
x
y
z
P
O
The relationship between the Cartesian coordinates P (x, y, z)and cylindrical coordinates (r, , z) is given by
x = r cos
y = r sin
z = zMath 55 Cylindrical & Spherical Coordinates 6/ 23
Cartesian Cylindrical
Remark
1 By the distance formula, r2 = x2 + y2.
2 dV = dz dA = rdz dr d
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Exercises
1 Write the following equations in cylindrical coordinates.
a. z =
3x2 +
3y2
b. x2 + y2 = 2y
c. z2 = 1 + x2 + y2
d. 2x2 + 2y2 + z2 = 4
2 A solid E lies within the cylinder x2 + y2 = 1, below theplane z = 4, and above the paraboloid z = 1 x2 y2.Find the volume of the solid E.
3 Evaluate
22
4y24y2
2x2+y2
xz dz dx dy
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Spherical Coordinates
Consider a point P in space. Let be the distance of P fromthe origin, be the same angle as in cylindrical coordinates, and be the angle between the positive z-axis and the line segment
OP .
x
y
z
P
O
The ordered triple (, , ) is called the spherical coordinates ofP .
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Surfaces in Spherical Coordinates
If = c, a constant then we have a sphere.
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Surfaces in Spherical Coordinates
If = c, a constant then we have a half-plane.
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Surfaces in Spherical Coordinates
If = c, a constant then we have a cone.
Figure: 0 c pi2 Figure: pi2 c pi
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Spherical CartesianSuppose P (x, y, z) has Spherical coordinates (, , ).
z
rx
y
P
Then z = cos. Let r be as in cylindrical coordinates. Thenr = sin. Therefore, x = r cos = sin cos andy = r sin = sin sin
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Spherical Cartesian
The relationship between the Cartesian coordinates P (x, y, z)and spherical coordinates (, , ) is given by
x = sin cos
y = sin sin
z = cos.
Also, the distance formula shows
2 = x2 + y2 + z2.
Note that > 0 and 0 pi.
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Spherical CartesianExample
Find the Cartesian coordinates of the point with sphericalcoordinates
(2, pi4 ,
pi3
).
Solution.
x = sin cos = 2(
sinpi
3
) (cos
pi
4
)= 2
(3
2
) (2
2
)=
6
2
y = sin sin = 2(
sinpi
3
)(sin
pi
4
)= 2
(3
2
)(2
2
)=
6
2
z = cos = 2(
cospi
3
)= 2
(1
2
)= 1.
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Spherical Cartesian
The point with spherical coordinates(2, pi4 ,
pi3
)is the point with
Cartesian coordinates (62 ,62 , 1).
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Spherical CartesianExample
A point P has Cartesian coordinates (0, 2
3,2). Find thespherical coordinates of P .
Solution. =
x2 + y2 + z2
=
0 + 12 + 4 =
16 = 4
cos =z
=24
= 12
=2pi
3,4pi
3
2pi
3
cos =x
sin= 0
=pi
2,3pi
2
pi
2because 0 pi since y > 0 Therefore, the sphericalcoordinates of P are (4, pi2 ,
2pi3 ).
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Triple Integrals in Spherical Coordinates
Suppose E is a solid given in spherical coordinates by
E = {(, , )|a b, , c d}
where a 0, 2pi and d c pi. We call this aspherical wedge.
We subdivide E into smaller sub-wedges Eijk by means ofequally spaced spheres = i, half-planes = j and half-cones = k.
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Triple Integrals in Spherical Coordinates
Consider the sub-wedge Eijk.
The volume of thesub-wedge can beapproximated by arectangular box withdimensions
, i, i sink
If Vijk is the volume ofEijk, then
Vijk 2i sink
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Triple Integrals in Spherical Coordinates
Suppose (i, j , k) has Cartesian coordinates (xijk, y
ijk, z
ijk).
ThenE
f(x, y, z)dV = liml,m,n
li=1
mj=1
nk=1
f(xijk, yijk, z
ijk)Vijk
which gives the formulaE
f(x, y, z)dV =
dc
ba
f( sin cos , sin sin , cos)2 sinddd
where E is the spherical wedge given by
E = {(, , )|a b, , c d}
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Triple Integrals in Spherical Coordinates
Example
Evaluate
E
z dV where E lies between the spheres
x2 + y2 + z2 = 1 and x2 + y2 + z2 = 4 in the first octant.
Solution. The equations of the spheres in sphericalcoordinates are = 1 and = 2. Since E is in the first octant,
E ={
(, , ) : 1 2, 0 pi2, 0 pi
2
}.
Hence,E
z dV =
pi2
0
pi2
0
21
( cos)2 sinddd
=
pi2
0
pi2
0
213 cos sinddd =
15pi
16
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Exercises
1 Write the following equations in spherical coordinates.
a. x = yb. z2 = x2 + y2
c. x2 + z2 = 9d. x2 + y2 + z2 = 2z
2 Evaluate
B
e(x2+y2+z2)
32 dV where B is the unit ball
centered at the origin.
3 Use spherical coordinates to find the volume of the solidthat lies above the cone z =
x2 + y2 and below the
sphere x2 + y2 + z2 = z.
4 Evaluate
30
9y20
18x2y2x2+y2
(x2 + y2 + z2) dz dx dy.
5 Find the mass of a solid E with constant density if E liesabove the cone z =
x2 + y2 and below the sphere
x2 + y2 + z2 = 1.
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References
1 Stewart, J., Calculus, Early Transcendentals, 6 ed., ThomsonBrooks/Cole, 2008
2 Dawkins, P., Calculus 3, online notes available atwww.lamar.com/pauldawkins
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