Question

whats the volume of the region bounded above by y=sqrt(x) and below by the line y=x, around the y axis

Answer 1

Using the shell method, the volume is given by the integral
\[V=2\pi\int_0^1x\big(\sqrt{x}-x\big)~dx\]
Here, \(x\) is the radius of each shell and \(\sqrt{x}-x\) is the height of each shell.