Volume obtained around $y=\sqrt{a}$

Question

Let $g$ be a function. $g(x)=\sqrt{x}-\sqrt{a}$. I want to find the volume of the region obtained rotating around $y=\sqrt{a}$. The picture is:

Thanks!

Answer 1

The distance from the axis of rotation to the curve is $r = 2\sqrt a - \sqrt x$. So a vertical slice of the solid of revolution, of thickness $dx$, will have area $\pi r^2dx = \pi(4a + x - 4\sqrt{ax})dx$. You just have to integrate this expression (over the interval $[0,4a]$, if that is what you want).