MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let U be the region in the plane bounded above by y=sinx, below by the x-axis, and on the left and right by x=0 and x=π, respectively. A solid has base U. For each x in [0,π], the cross section taken perpendicular to the x-axis is a semicircle whose diameter lies entirely within U.
Address the following prompts.
(a) Show that the volume W of the solid can be expressed as a definite integral involving an elementary function. Clearly identify the integrand and limits of integration, providing a complete derivation.
(b) Evaluate the integral obtained in part (a) to find the exact volume of the solid.
(c) Suppose instead the cross sections were full circles (with the same diameters as in part (a)). Without re-computing any integrals, determine the new volume and explain your reasoning.