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Math 251-copyright Joe Kahlig, 15C
Section 13.9/13.10: Spherical Coordinates and Triple Integrals
Spherical Coordinates:
A Cartesian point (x, y, z) is represented by (ρ, θ, φ) in the Spherical Coordinate System where ρ ≥ 0
and 0 ≤ φ ≤ π.
Example: Find the spherical coordinates for the points (−1,
Example: Convert the equations to spherical coordinates.
A) x2 + y 2 + z 2 = 25
B) z = 12 − 4x2 − 4y 2
C) z =
p
3x2 + 3y 2
√
3, 2) and (−1,
√
3, −2)
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Math 251-copyright Joe Kahlig, 15C
Triple Integrals in Spherical Coordinates
In this coordinate system, the equivalent of a box is a spherical wedge
E = {(ρ, θ, φ)|a ≤ ρ ≤ b, α ≤ θ ≤ β, c ≤ φ ≤ d}
where a ≥ 0, β − α ≤ 2π, and d − c ≤ π
Zd Zβ Zb
ZZZ
f (x, y, z)dV =
E
c
α
f (ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ) ρ2 sin φ dρdθdφ
a
Note: Spherical coordinates are used in triple integrals when surfaces such as cones and spheres form
the boundary of the region.
Z3
√
Z9−x2
√
2
2
9−x
Z −y
Example: Evaluate
√
−3 − 9−x2
0
z2
p
x2 + y 2 + z 2 dzdydx
Math 251-copyright Joe Kahlig, 15C
Example: Find the volume of the solid that lies above the cone z =
x2 + y 2 + z 2 = z.
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p
x2 + y 2 and below the sphere
Example: Set of the triple integral that will find the volume of the smaller wedge cut from a sphere
π
of radius 3 by two planes intersecting along a diameter of the sphere at an angle of .
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Math 251-copyright Joe Kahlig, 15C
Example Find the volume that is inside the sphere x2 + y 2 + z 2 = 4 and above the plane z =
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√
3.