Multivariate Calculus - Bruce E. Shapiro

166 LECTURE 20. TRIPLE INTEGRALS
Triple Integrals in Cylindrical Coordinates
Recall the conversion between Cartesian and cylindrical coordinates: to find (x, y, z)
given (r, θ, z),
x = r cos θ, y = r sin θ
with z unchanged; to find (r, θ, z) given (x, y, z)
r 2 = x 2 + y 2 ,
, tan θ = y/x
again with z unchanged. We observe that cylindrical coordinates are identical to
polar coordinates in the xy-plane, and identical to Cartesian coordinates in the z-
direction. We can fill up volume V with micro-volumes that have bases given by
the volume element in polar coordinates
and height dz, hence
and
dA = rdrdθ
dV = dA × dz = rdrdθdz
f(x, y, z)dV = f(r cos θ, r sin θ, z) rdr dθ dz
V
V
Example 20.4 Use cylindrical coordinates to find the volume of solid bounded by
the paraboloid
z = 9 − x 2 − y 2
and the xy-plane.
Solution. The paraboloid intersects the xy-plane in a circle of radius 3 centered
about the origin.In cylindrical coordinates,
z = 9 − x 2 − y 2 = 9 − r 2
and therefore
V =
=
=
=
=
∫ 2π ∫ 3 ∫ 9−r 2
0 0
∫ 2π ∫ 3
0 0
∫ 2π ∫ 3
0
∫ 2π
0
= 81π
4
0
0
dzrdrdθ
(9 − r 2 )rdrdθ
(9r − r 3 )drdθ
( 9
2 r2 − 1 ) 3
4 r4 dθ
0
( 81
2 − 81
4
)
(2π)
Revised December 6, 2006. Math 250, Fall 2006