Find the maximum volume of a cylinder

Question

In order to inscribe a rectangle between the parabola $$ y=-x^2+2 $$and the x-axis so that the maximum cylinder volume is obtained by a complete rotation around the y-axis, I know that I should derivate the formula of the volume in order to find the maximum. However, I think I'm missing something since the result isn't what I expected (result: y=1).

Answer 1

The volume of a cylinder is given by $V = r^{2}*h*pi$. In your situation $h=y$ and $r=x$. The problem at hand is:

$$ max \ f(x,y) = yx^{2}\pi \\ subject \ to: \ y=-x^{2} +2, \quad x \ge 0\\$$

Combine equality constraint with cost function: $f(y) = 2\pi y - y^{2}\pi $. $ f'(y) = 2\pi - 2 \pi y$, equals zero if y = 1.