You're given your z limits:

Looking at the cross section of the paraboloid at , we see that we have a circle of radius 1. This means that and

Thus, our triple integral is

The reason why its multiplied by for is because of how I defined my x and y boundaries. It took into consideration only 1/4th of the region , so in order to get the correct value after integrating it, you need to quadruple the value...

Does this make a little more sense?

--Chris