Volume rotation basic calculus answer doesn’t match up

Question

Volume solid generated by rotating the region in the second quadrant that lies between the curves $x=-y$ and $y=x^2$ around the $x$ axis. I am confident of my calculus skills but the answer I get does not match with the source that I am using. Please calculate the volume and tell me the answer.

Answer 1

We are going to find the volume generated by the line first and then the volume generated by the quadratic since the line lies on top of the quadratic in our desired region:

$\int\ \pi(-x)^2\ dx$ Where our region is $[-1,0]$ (Equals $ \frac{\pi}{3}$

$\int\ \pi(x^2)^2\ dx$ Over the same region. (Equals $\frac{\pi}{5}

Since our volume lies in between those regions, we will subtract them and be done: $\frac{\pi}{3}-\frac{\pi}{5}=\frac{2\pi}{15}$

This method works as we are calculating the radius of our solid instantaneously and using it find the area of the circle and summing these circles will give us the volume of our circle.