2
$\begingroup$


I have seen a lot of exercises where they solve a triple integral using spherical coordinates. But I'm confused about the limits that one should use. For example when they integrate over a sphere like $x^2 + y^2 + z^2 = 4$ I do understand why the limit are $0 \leq \rho \leq 2$ , $0\leq \theta \leq 2\pi$, but I can't get why this one ends in $\pi$ and not in $2\pi$, $0 \leq \phi \leq \pi$.
Thank You!

$\endgroup$
1
$\begingroup$

$0 \leq \phi \leq \pi$ Otherwhise you would be counting the volume twice!

Indeed, fix $\ \theta$ then, the equations $0 \leq \rho \leq 2$ , $0\leq \phi \leq \pi$ describe a half circle that has "vertical" diameter (see picture below)

By making this half-circle turn around $0\leq \theta \leq 2\pi$ we get the full sphere

enter image description here

$\endgroup$
  • $\begingroup$ thanks a lot. the half-circle (0 to π) definition for ϕ and turning it around through 360 degrees to cover the entire θ range (0 to 2π) is the best explanation of this problem i've also been suffering with for many many years. $\endgroup$ – nyxee Feb 21 '18 at 21:05
1
$\begingroup$

Stand with your arm held directly above your head. Pretend your arm has radius $\rho=2$. Now swing it through $0\le\phi\le\pi$. You ought to have made a semicircle, and now your arm is resting against your leg. Next, keep swinging your arm through $\phi$, but also turn full circle on the balls of your feet, $0\le\theta\le2\pi$. Your arm ought to have swept out a sphere.

If your arm had initially gone from $0\le\phi\le2\pi$, you would have swept out two spheres in the end.

$\endgroup$
0
$\begingroup$

Using Mathematica, i have noticed that when we interchange the limits, the volume doesn't change.. i.e using { 0≤ϕ≤π and 0≤θ≤2π } OR { 0≤ϕ≤2π and 0≤θ≤π } yielded the same volume

SphericalPlot3D[1, {\[Theta], 0, 3 Pi/2}, {\[Phi], 0, Pi}, AxesLabel -> {x, y}, PlotLabel -> r == 1, AxesEdge -> {{-1, -1}, {1, -1}, {-1, -1}}]
SphericalPlot3D[1, {\[Theta], 0, Pi}, {\[Phi], 0, 3 Pi/2}, AxesLabel -> {x, y}, PlotLabel -> r == 1, AxesEdge -> {{-1, -1}, {1, -1}, {-1, -1}}]
$\endgroup$

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.