Use cylindrical coordinates. find the volume of the solid that lies within both the cylinder x2 + y2 = 25 and the sphere x2 + y2 + z2 = 64.

The volume will consist of the cylinder up to where it intersects with the sphere plus the two end caps (top/bottom parts of sphere)

First rewrite equations using cylindrical coordinates:


Find intersection points:
Sub in r =5 into sphere equation


Set up integral:
The general volume for cylinder is 

For the inner cylinder you have: 

It works out the same as cylinder formula 

For the end caps of sphere, use the triple integral for cylinder volume, however change the limits for 'r' in terms of 'z'.
This is because the radius will be shrinking as 'z' value increases.
we know  from sphere equation.
Also top/bottom will have equal volumes so you can just double the integral to get volume of both sections.



Combining the volumes yields: