Here's the question $$\iiint x^2 \,dx\,dy\,dz.$$ I am asked to evaluate this integral over the region $$D:=\left \{ (x,y,z) \in\mathbb{R}^3 :x^2 \leq y^2+z^2 \leq 4\right \}.$$

I've shown that :

  • $ 2 \leq x \leq -2 $


$$\iiint x^2 \,dx\,dy\,dz= \int_{-2}^{2}dx \int\int x^2 dydz$$

From here,do i have to use the cylindrical coordinates?

Any support for this question would be appreciated.

  • $\begingroup$ ok @Arthur i try to do it $\endgroup$ – andrew Jan 23 at 18:21
  • 1
    $\begingroup$ The region is a (sort-of) hollowed-out cylinder centered around the $x$-axis. Maybe cylindrical coordinates makes this easier? (Sorry about saying spherical, I misread the inequalities.) And if you want to do it in Cartesian coordinates, I suspect that you want $x$ to be the innermost integral. $\endgroup$ – Arthur Jan 23 at 18:22
  • $\begingroup$ yeah, you're right $\endgroup$ – andrew Jan 23 at 18:23
  • $\begingroup$ i obtained that the solution is $\frac{64*\pi}{3}$ $\endgroup$ – andrew Jan 23 at 18:36

I use cylindrical coordinates $y^2+z^2=r^2$ and $x=h$. Then $h$ varies between $-r$ and $r$. Then my integral is $$\iiint_D rdr d\theta h^2dh=\int_0^{2\pi}d\theta\int_0^2dr\ r\int_{-r}^rh^2dh$$


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.