Currently I am working on this problem that requires me to calculate this triple integral when I am given cone/plane intersection. The issue is that integrals in both cases (when using spherical and cylindrical coordinates) lead to, even more, complicated ones. Only hints required. Thank You. enter image description here

Calculate $$\int \int \int_E \sqrt{x^2+y^2+z^2}\space dV$$

where $E$ is the is region bound by the plane $z=3$ and the cone $z=\sqrt{x^2+y^2}$

  • $\begingroup$ So you're finding the volume of the cone? $\endgroup$ – Andrew Li Jun 3 '18 at 2:42
  • $\begingroup$ What did you get when you converted to cylindrical coordinates? $\endgroup$ – Matthew Leingang Jun 3 '18 at 2:47
  • $\begingroup$ This should work out quite nicely in either cylindrical or spherical coordinates. You had better check your work. @AndrewLi: No, this not the volume; it's the total mass if the density is given by the integrand. $\endgroup$ – Ted Shifrin Jun 3 '18 at 2:47
  • $\begingroup$ @TedShifrin Ok, what I thought. I was unsure OP never mentioned density (even though they put the function in their post), just wanted to confirm $\endgroup$ – Andrew Li Jun 3 '18 at 2:50
  • $\begingroup$ When I got to calculating dr part intervals gave me 1/0 at some point. $\endgroup$ – kenobe Jun 3 '18 at 2:52

In cylindrical coordinates, the cone has equation $z=r$; it intersects the plane in a circle of radius $3$. So the integral is equal to $$ \int_0^{2\pi} \int_0^3 \int_r^3 \sqrt{r^2+z^2}r\,dz\,dr\,d\theta = 2\pi \int_0^3 \int_r^3 \sqrt{r^2+z^2}r\,dz\,dr $$ As written the integral on the right needs a trigonometric substitution; it would be easier to rearrange the limits: $$ 2\pi \int_0^3 \int_r^3 \sqrt{r^2+z^2}r\,dz\,dr = 2\pi \int_0^3 \int_0^z \sqrt{r^2+z^2}r\,dr\,dz $$ Can you take it from there?

  • $\begingroup$ Yes I can. Thank you :) $\endgroup$ – kenobe Jun 3 '18 at 3:04

When you intersect the cone and the plain you get a circle. The region of integration is the interior of that circle. Watch for the limits of the integration. You want to describe the volume between $z=3$ and $z=\sqrt {x^2+y^2}$ nothing more and nothing less. Drawing a graph is very helpful to find the correct limits.


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