# 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.

• I'm pretty sure it's the same. I'm taking the arc in the 1st quadrant, rotating it around the x-axis, then doubling the result. MIT Prof takes the arc in the 1st+4th quadrants and rotates it around the y-axis, and then doesn't need to double. – user1956609 Aug 22 '13 at 1:28
• That's a small difference of strategy. I prefer yours, like to avoid negative numbers if I can. But a prolate spheroid and an oblate spheroid (same pair of "diameters") have different volumes. – André Nicolas Aug 22 '13 at 1:33
• OHHHH. Okay I see it now. I created something shaped like a pill, while MIT Prof wanted something shaped like a tire. Never occurred to me that rotations can give such different answers. Thanks! – user1956609 Aug 22 '13 at 1:38
• You are welcome. In general, rotating about different lines yields different shapes and different volumes. – André Nicolas Aug 22 '13 at 1:46

Your own attempt is correct , but the article wanted to do differently as @Andre pointed. This could be useful if you used the following codes which is so easy to work in Maple's environment:

  [> with(Student[Calculus1]):
[> VolumeOfRevolutionTutor();


It make you to work with your function (explicitly defined) in a interactive frame. After you inserted the above codes, the following frame appears:

This is the region you did by yourself. Try to use it! It's so fun! (-:

• Awesome animation! – Namaste Aug 23 '13 at 0:37