Question

The base of a solid is the region enclosed by y=x^3 and the x-axis on the interval [0,4]. Cross sections perpendicular to the x-axis are semicircles with diameter in the plain of the base. Write an integral that represents the volume of the solid.

I drew a picture, but I learned how to do this so long ago (in calc AB) that I don't remember how to start.

Answer 1

So, add up all the little wafers with diameter x^3, perpendicular to the x-y plane.

They would have their centers at (x,y/2) with radius y/2 = x^3/2. The volume of a small semi-circular wafer of radius r and thickness dx is 1/2 πr^2 dx

v = ∫[0,4] 1/2 π(x^3/2)^2 dx
= π/8 ∫[0,4] x^6 dx
...