Question

3) The base of a solid is the first quadrant region between the curve y = 2 ⋅ sin ⁡ xand the x-axis on the interval [ 0 , π ]. Cross sections perpendicular to the x-axis are semi circles with diameters in the x-y plane. Sketch the solid. Find the volume of the solid.

Answer 1

Let us draw given region

Cross section perpendicular to the x axis are circles with diameter in xy plane

So height y is the diameter

Volume is given by the integral

Area of circle with diameter y is given by

But we have y=2sinx

X varies from 0 to π

So volume is