Problems: Spherical Coordinates
1.√ Find the volume of a solid spherical cap obtained by slicing a solid sphere of radius
a 2 by a plane at a distance a from the center. (See picture.)
Answer: In session 76 we found the limits:
√
inner ρ: a/ cos φ to a 2,
middle φ: 0 to π/4,
outer θ: 0 to 2π.
2π
π/4
a√2
ρ2 sin φ dρ dφ dθ.
Volume =
0
0
a/ cos φ
√
√
a 2
1
3
2a3 2
a3 sin φ
=
sin φ −
.
Inner:
ρ sin φ 3
3 cos3 φ
3
a/ cos φ
√
√
√
π/4
2a3 2
a3
2a3 a3
2 2a3 a3
2 2a3 5a3
Middle: −
cos φ −
=−
−
− (−
− )=
−
.
3
6 cos2 φ
3
3
3
6
3
6
φ=0
√
2 2a3 5a3
a3 π √
Outer: 2π(
−
)=
(4 2 − 5) ≈ 0.7a3 .
3
6
3
The volume of the entire sphere is about 12a3 and we’re looking at approximately the top
sixth of its height. The sphere has more volume near its midpoint than at its top and
bottom, so this answer seems reasonable.
MIT OpenCourseWare
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18.02SC Multivariable Calculus
Fall 2010
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