Science:Math Exam Resources/Courses/MATH101/April 2007/Question 02 (c)/Solution 1

A picture is included below.

Imagine yourself standing on the x-axis at x=-1 looking towards the rotated object. This top down view (the circles in the picture) show that what we are doing is taking a cylinder of radius 1 and height $\pi$, and subtracting from it the shape that we get from removing the inner shape of radius $(1-\sin(x))^{2}$. This gives the volume to be

$V=\int _{0}^{\pi }\pi (1^{2}-(1-\sin(x))^{2})\,dx=\pi \int _{0}^{\pi }(2\sin(x)-\sin ^{2}(x))\,dx$

completing the question.