Asolid has its base in the xy-plane, bounded by the x axis, the y axis and the function y=3-x⁵. if cross sections are taken perpendicular to the x axis are semicircles whose diameters are in the xy-plane, what is the volume of this solid?

Attached is a plot of the base with one of the cross sections (at ).

The area of any one cross section is given by , where is the radius of the circular cross section. In terms of the sections' diameters , the area would be .

Each section's diameter is determined by the vertical distance (in the x-y plane) between the curve and the x-axis (), or simply . So the area of any one cross-section for a given is .

The region extends from to (the positive root of ), so the volume of the solid would be



You can compute this by expanding the integrand, then integrating term by term. You should find a volume of .

The clear answer is clearly 25 is very clear to me