Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Use cylindrical shells to find the volume $V$ of the solid torus (the donut-shaped solid shown in the figure) with radii $r$ and $R$.

This is as far I have come. How can I solve further?

share|cite|improve this question
That last integral smells like trig substitution. Try something like $t = r\cos\theta$. – Neal Jan 25 '13 at 2:49
This is a lot easier with Pappus Throrem, volume=area*length of path of centroid. – Maesumi Jan 25 '13 at 3:26
up vote 1 down vote accepted

To solve $\int_{-r}^r\sqrt{r^2-t^2}dt$, you can use the substitution $t=r\sin\theta$. Then if $t=r$, $\theta=\pi/2$; $t=-r$, $\theta=-\pi/2$. Also, $dt=r\cos\theta d\theta$ and $\sqrt{r^2-t^2}=r\cos\theta$. Hence, we have $$\int_{-r}^r\sqrt{r^2-t^2}dt=\int_{-\pi/2}^{\pi/2}r^2\cos^2\theta d\theta =r^2\int_{-\pi/2}^{\pi/2}\cos^2\theta d\theta.$$ I will leave the remaining part to you.

share|cite|improve this answer
Thank you! That helped. :) – None Jan 25 '13 at 2:56

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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