Volume generated by rotating axis $X =2$ and the lines $x =0$ and $y = 1$

Question

The question

I wrote the shaded area equation and add 2 to it because it has shifted two units to the right, however the answer is wrong, where did I go wrong?

Answer 1

Your error is in assuming the general equation for the volume of rotation to be

$$ V=\pi\int y^2dx $$

but in fact, for rotation about a vertical axis it should be

$$ V=2\pi\int xydx $$

If you make the substitution $X=x-2$ and apply the above equation you will have

$$ V=2\pi\int_0^1 X\sqrt{1-X}\ dX $$

I will leave it to you to show that the correct answer is $V=8\pi/15$.