0
$\begingroup$

Question: Find the volume of the region that is contained by the cylinder $x^2+y^2=81$, bounded above by $z=x$ and below by the $xy$-plane.

I have tried the integral $\int_{0}^{2pi}\int_{0}^{9}\int_{0}^{rcos\theta}r^3dzdrd\theta$ and got $0$, same when $0\leq\theta\leq\pi$. Then I tried $\int_{0}^{pi/2}\int_{0}^{9}\int_{0}^{rcos\theta}r^3dzdrd\theta$ and got 11809.8. Which is also wrong.

$\endgroup$
  • $\begingroup$ why is there $r^3$? shouldnt there just be an r in the change of coordinates , im assuming your integrating with the caratheristic function. $\endgroup$ – Someone Apr 16 '19 at 22:46
2
$\begingroup$

If you are bounded below by the $xy$ plane and bounded above by the $z=x$ plane, then $z\geqslant0$ and $z\leqslant x$. So, $x\geqslant 0$ and so, since $x=\rho\cos\theta$, $\theta\in\left[-\frac\pi2,\frac\pi2\right]$. So, you consider the integral$$\int_{-\frac\pi2}^{\frac\pi2}\int_0^9\int_0^{\rho\cos\theta}\rho\,\mathrm dz\,\mathrm d\rho\,\mathrm d\theta.$$You should get $486$.

$\endgroup$

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.