Lecture 16, October 31
• Triple integrals in cylindrical coordinates. One has the formulas
x = r cos θ,
x2 + y 2 = r 2 ,
y = r sin θ,
dV = r dz dr dθ.
• Triple integrals in spherical coordinates. One has the formulas
x = ρ sin ϕ cos θ,
y = ρ sin ϕ sin θ,
dV = ρ2 sin ϕ dρ dϕ dθ.
z = ρ cos ϕ,
.....................................................................................
√
Example 1. Consider the solid G which is bounded by the cone z = x2 + y 2 from
below and by the sphere x2 + y 2 + z 2 = 8 from above. To find its projection R onto
the xy-plane, we find the points at which the cone meets the sphere, namely
x2 + y 2 + z 2 = 8
x2 + y 2 + (x2 + y 2 ) = 8
=⇒
=⇒
x2 + y 2 = 4.
This gives a circle of radius 2 in the xy-plane, so the volume of G is
∫ √
∫∫∫
∫ ∫ √
Volume =
2
4−x2
−2
√
− 4−x2
dV =
G
8−x2 −y 2
√
dz dy dx.
x2 +y 2
In terms of cylindrical coordinates, the circle x2 + y 2 = 4 becomes r2 = 4 and so
∫
2π
∫
2
∫
√
8−r2
Volume =
r dz dr dθ.
0
0
r
In terms of spherical coordinates, finally, the equation of the cone is
√
r
z = x2 + y 2 = r =⇒ tan ϕ = = 1 =⇒ ϕ = π/4
z
and we can compute the volume of the solid as
∫
2π
∫
π/4
∫
√
∫
8
2π
∫
ρ2 sin ϕ dρ dϕ dθ =
Volume =
0
∫
0
2π
=
0
∫
2π
=
0
0
0
0
π/4
[
3
ρ sin ϕ
3
]π/4
√
√
∫ 2π [
16 2
16 2
sin ϕ dϕ dθ =
−
cos ϕ
3
3
0
0
√ (
√ )
16 2
2
32π √
· ( 2 − 1).
1−
dθ =
3
2
3
∫
π/4
]√8
dϕ dθ
ρ=0
dθ
ϕ=0