Triple Integrals in Cylindrical Coordinates

Many applications involve densities for solids that are best expressed in non-Cartesian coordinate systems. In particular, there are many applications in which the use of triple integrals is more natural in either cylindrical or spherical coordinates.

For example, suppose that f( r,q) ³ g( r,q) in polar coordinates and that U( x,y,z) is a continuous function. If S is the solid between z = f( x,y) and z = g( x,y) over a region R in the xy-plane , then

ó õ ó õ ó õ S  U( x,y,z) dV =   ó õ ó õ R  é ë ó õ f( x,y) g(x,y)   U( x,y,z) dz ù û   dA

Let's suppose now that R is bounded in polar coordinates by q = a, q = b, r = p( q) , and r = q(q) . Since dA = rdrdq in polar coordinates, a change of variables into cylindrical coordinates is given by

ó õ ó õ ó õ S  U( x,y,z) dV = ó õ b a  ó õ q( q) p( q)   ó õ g( r,q) f(r,q)   U( rcosq,rsin( q) ,z)   r  dz dr dq

(1)

In practice, however, it is often more straightforward to simply evaluate the first integral in z and then transform the resulting double integral into polar coordinates.              

EXAMPLE 1    Evaluate the following

ó õ 1 0  ó õ Ö 1-x2    0 ó õ 2xy 0    (x2+y2) dz dy dx

Solution: Rather than employ (1) directly, let's first evaluate the integral in z. That is,

ó õ 1 0  ó õ Ö 1-x2    0 ó õ 2xy 0    (x2+y2) dz dy dx = ó õ 1 0  ó õ Ö 1-x2    0   ( zx2+zy2 )  ê ê 2xy 0 dy dx = ó õ 1 0  ó õ Ö 1-x2    0   2xy( x2+y2) dy dx = ó õ ó õ R  2xy( x2+y2) dA

where R is the quarter of the unit circle in the 1st quadrant. In polar coordinates, R is bounded by q = 0, q = p/2, r = 0, and r = 1. Thus,

ó õ 1 0  ó õ Ö 1-x2    0 ó õ 2xy 0    (x2+y2) dzdydx = ó õ p/2 0  ó õ 1 0  2rcos(q) rsin( q) r2  rdrdq = ó õ p/2 0  ó õ 1 0  sin( 2q) r5  drdq = ó õ p/2 0  sin( 2q)  r6 6 ê ê 1 0  dq = ó õ p/2 0   1 6 sin( 2q) dq =  1 6