I am having trouble setting up triple integrals to find a volume of a given solid. Here is one of the questions with which I am having trouble.

Now I can see that the projection of the solid on the xy plane is the circle x^2 + y^2 = 9. And I think I can visualize the plane z = y + 3 with respect to the cylinder - it slices the cylinder in half diagonally. But I am not sure how to set up the triple integral.

This is the way I would set up the integral

Let E be the solid in question. Then

E = {(x,y,z) | -3 <= x <= 3, -(9-x^2)^1/2 <= y <= (9-x^2)^1/2, 0 <= z <= y + 3 }

If I was to integrate using these limits I would then integrate with respect to z first, then y, then x.

Now I know that it would be easier to eventually convert to polar coordinates but I would like to know if the way I set up the triple integral to find the volume of the solid is correct so far.

Now I should integrate with respect to x. But I can't seem to do it / remember how to integrate this part:

int.[-3,3] 6(9-x^2)^1/2 dx

Any further help is appreciated.

BTW, would I be able to integrate this integral with trig substitution? If so. I cannot remember how to do that. Could someone guide me / help me recall the process? Thanks again.