You don't need to integrate anything.
First we want the volume of a sphere, which is:
We are tasked with finding a rate of change, which means we are going to be dealing with derivatives.
We are told that is equal to "10 cubic feet per second". We need to find .
Taking the derivative of the volume of a sphere, and differentiating it with respect to time, will yield us an equation, that we can solve for .
If you need further help lemme know.