Question

The base of a solid in the xy-plane is the first-quadrant region bounded y = x and y = x2. Cross sections of the solid perpendicular to the x-axis are semicircles. What is the volume, in cubic units, of the solid?

Answer 1

the area of the cross section is \(A(x) = \dfrac{1}{2} \times \dfrac{\pi D^2}{4} = \dfrac{1}{2} \times \dfrac{\pi (x - x^2)^2}{4} = \dfrac{\pi (x - x^2)^2}{8}\)

and so the volume should be summat like: \(\frac{\pi}{8}\int\limits_0^1 ~ (x - x^2)^2 ~ ~ dx \)