Evaluating Triple Integrals with Cylindrical Coordinates

Suppose that E is a type 1 region whose projection D onto the xy-plane is conveniently described in polar coordinates see Figure .

In particular, suppose that is continuous and 

where D is given in polar coordinates by

we obtain

The formula for triple integration in cylindrical coordinates. It says that we convert a triple integral from rectangular to cylindrical coordinates by writing , leaving z as it is, using the appropriate limits of integration forand , and replacing  by . It is worthwhile to use this formula when E is a solid region easily described in cylindrical coordinates, and especially when the function  involves the expression .

EXAMPLE 2:  A solid E lies within the cylinder , below the plane , and above the paraboloid . See Figure below the density at any point is proportional to its distance from the axis of the cylinder. Find the mass of E

In cylindrical coordinates the cylinder is and the paraboloid is , so we can write

Since the density at  is proportional to the distance from the z-axis, the density function is

where K is the proportionality constant. Therefore, the mass of E is

EXAMPLE 3:  Evaluate

This iterated integral is a triple integral over the solid region

and the projection of  E onto the xy-plane is the disk . The lower surface of E is the cone  and its upper surface is the plane . This region has a much simpler description in cylindrical coordinates: 

Therefore we have