How to find the volume using integrals

Question

Find the volume using integrals, that is formed/bounded with lines $y^2 + x - 4 = 0, x = 0$, which rotates around $y-axis$.

I tried to solve it by visualizing the model from top-down view, but that didn't get me anywhere(I wasn't sure how to find it through integrals). How can I find the volume?

Answer 1

Volume of revolution is $$\int_a^b \pi x^2 dy$$ Find the coordinates of intersection of the parabola with the y-axis, these are your $b$ and $a$ values.

Then rewrite the equation of $x$ in terms of $y$, and square the result to get $x^2$ in terms of $y$

Plug the rest into the formula and you are done.

Answer 2

This is what you did, right?

$\int_{-2}^{2} \pi(4-y^2)^2dy$

The answer should be $107.233$