I'm working on trying to get a 3D ellipsoid from the area bounded by two curves. To get this curve, I am reflecting the parabola $\displaystyle y = -x^2 + x + 2$ over the x-axis to get $\displaystyle y = x^2 - x - 2$. This gives me an area bounded on the x-interval (-1, 2). So when I set up my bounds function, I get $\displaystyle Bounds = -2x^2 + 2x + 4$.

From here, I want to revolve this area around the x-axis to give me a 3D ellipsoid. I am debating between either integrating this bounds function I have come up with from -1 to 2 wrt x; or integrating the area of an ellipse $\displaystyle y = \pi * A * B$, where A and B are the radii across the major and minor axes respectively, using the same limits of integration and wrt x. I am thinking that integrating the Bounds function twice will give me volume. Is this correct? Could I (or should I) instead Integrate $\displaystyle Bounds^2$ to get the volume?

For part two of my problem, I want to be able to find the area of a slice of the ellipsoid at any point along the z-axis. I know that as I travel along the z-axis, the size of my slice will change. However, beyond that, I'm not exactly sure how to extrapolate this area. Can anyone guide me on how to do this?

Thanks in advance for everyone's help!

For what it's worth, to put everything in context, I am working on the math to model 3D gravity in the context of bounding the space from start to finish for a character jumping.