What is the volume of this object using calculus?

Question

I have been trying to solve this problem and I have yet to figure it out.

A volume is described as follows:

1.) The base is the region bounded by x = -y² + 12y + 20 and x = y² - 26y + 176.

2.) Every cross section perpendicular to the y-axis is a semi-circle.

Find the volume of this object.

How would you go about finding the volume?

Answer 1

The first thing to do is find where the two curves intersect. So -y² +12y +20 = y² -26y +176 this gives y=6,13. To find the diameter D of the semicircle subtract one curve equation from the other to get 2y² -38y + 156 and the radius is D/2. So the radius is y² -19y +78 and the area is πr²/2 for the semicircle. which gives π

π(y² -19y +78)²/2. Volume V = π/2 ∫₆¹³ (y² -19y +78)²dy. Doing the integration will give the volume.