Volume of a spheroid using calculus MIT's online Calc course includes this problem, where we're asked use integration along with a bound region in 2d space to find the volume of a spheroid. I understand the solution given by the professor, but I originally found my answer by rotating over the x-axis (the professor's solution rotates over the y-axis). I keep getting a different answer and I can't figure out why. Here's my math: $$radius: y^2 = 1 - \frac{x^2}{4}$$ This produces the following integral: $$\pi \int_0^2 1 - \frac{x^2}{4} dx = \frac{4}{3}\pi$$ This should give the area for half the spheroid, so my final answer was $2*\frac{4}{3}\pi = \frac{8}{3}\pi$. The correct answer is $\frac{16}{3}\pi$. I've checked through this and can't see my error.

The original question asks for rotation about the $y$-axis. You are rotating about the $x$-axis.

That yields a different solid, with a different volume. The calculation you made for **your** solid is correct.