# Challenge

A student was asked to find the volume of the bowl-shaped figure generated when the curve y = x2 from x = 0 to x = 2 is revolved around the y-axis. She used the disk method and found the volume to be $\displaystyle \int_{0}^{4}{\pi ydy}=8\pi$. To check her work she used the method of cylindrical shells and found the same answer: $\displaystyle\int_{0}^{2}{2\pi x\left( 4-{{x}^{2}} \right)dx}=8\pi$.

The second part of the questioned asked for the value of x for which the bowl would be half full. So she first solved the equation $\displaystyle \int_{0}^{k}{\pi ydy=\frac{1}{2}\cdot 8\pi }$  and found $k=2\sqrt{2}$. This is a y-value so the corresponding x-value is $\sqrt{2\sqrt{2}}\approx 1.682$. She again checked her work by shells by solving $\displaystyle \int_{0}^{k}{2\pi x\left( 4-{{x}^{2}} \right)dx=\frac{1}{2}\cdot 8\pi }$ and found that $k=\sqrt{-2\left( \sqrt{2}-2 \right)}\approx 1.082$. (This is the x-value.)

Both computations are correct. Can you explain to her why her answers are different? Use the comment box below to share your explanations. I will post mine in a week or so.