The base of S is a circular disk (most would call this a circle!) with radius r. Parallel cross sections perpendicular to the base are isosceles triangles with height h and unequal side of the base.

a) Set up the integral for the volume of S

b) By interpreting the integral as an area, find the volume of A.

So, I took some mushrooms and now I think that I have this visualized (more or less). The base of the triangle straddles the x-axis, so its base is 2y and its height is h. If each end of the base touches the circle, then

So,

And,

Am I right so far? What I don't see is how to integrate this. I can do a trig substitution for the radical, but what do I do with h? I think that this is a clue that I have done something wrong.

Can anyone help? Thanks.