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.

  • $\begingroup$ 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. $\endgroup$ – user1956609 Aug 22 '13 at 1:28
  • $\begingroup$ 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. $\endgroup$ – André Nicolas Aug 22 '13 at 1:33
  • 1
    $\begingroup$ 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! $\endgroup$ – user1956609 Aug 22 '13 at 1:38
  • $\begingroup$ You are welcome. In general, rotating about different lines yields different shapes and different volumes. $\endgroup$ – 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:

enter image description here

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

enter image description here

  • $\begingroup$ Awesome animation! $\endgroup$ – Namaste Aug 23 '13 at 0:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.