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Evaluate the triple integral of: $f(x,y,z)=z(x^2+y^2+z^2)^{-3/2}$

Over the part of the ball: $x^2+y^2+z^2\le 16$ with $z\ge 2$

So I converted to spherical coordinates and got: $$f(\rho,\phi,\theta)=\rho cos(\phi)\rho^{-3/2}$$ $$f(\rho,\phi,\theta)=cos(\phi)\rho^{-1/2}$$

Then for the bounds: $$\rho^2\le 16$$ $$\rho\le 4$$ and $$\rho cos(\phi)\ge 2$$ $$2/cos(\phi)\le \rho$$

So: $$2/cos(\phi)\le \rho\le 4$$

Then since the plane crosses the sphere at $z=2$ where $\rho=4$, and $z=\rho cos(\phi)$: $$4cos(\phi)=2$$ $$\phi = \pi/3$$ So: $$0\le\phi\le\pi/3$$ Since it is a sphere I have: $$0\le\theta\le 2\pi$$

So my bounds are: $$2/cos(\phi)\le \rho\le 4$$ $$2/cos(\phi)\le \rho\le 4$$ $$0\le\theta\le 2\pi$$ Over: $$f(\rho,\phi,\theta)=cos(\phi)\rho^{-1/2}$$

With $dV=\rho^2 sin(\phi)d\rho d\phi d\theta$, I write the integral as:

$$\int_0^{2\pi}\int_0^{\pi/3}\int_{2/cos(\phi)}^4 \rho^{-1/2} cos(\phi) \rho^2 sin(\phi) d\rho d\phi d\theta$$

$$\int_0^{2\pi}\int_0^{\pi/3}\int_{2/cos(\phi)}^4 \rho^{3/2} cos(\phi) sin(\phi) d\rho d\phi d\theta$$

Is this all correct?

MathMan08
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1 Answers1

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Alternatively, you could set this up in cylindrical coordinates. The integral becomes

$$2 \pi \int_2^4 dz \, z \: \int_0^{\sqrt{16-z^2}} d\rho \, \rho\, (z^2+\rho^2)^{-3/2}$$

The result I get is $\pi$.

Ron Gordon
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  • How would you solve that? – MathMan08 Apr 06 '13 at 22:56
  • I did solve it. Work right to left. The inner integral may be done by substituting $u=\rho^2$. The outer integral is then very simple. – Ron Gordon Apr 06 '13 at 22:57
  • The volume of the sphere before being cut is $4/3 \pi r^3=4/3 \pi *4^3=256 \pi/3$. So we know the top half of the sphere has a volume of half that which is $128 \pi/3$. So $\pi/2$ for that section of the top half seems very small... – MathMan08 Apr 06 '13 at 23:01
  • Yeah, but you are not finding a volume, but integrating some function over that volume. – Ron Gordon Apr 06 '13 at 23:03
  • Oh, forgot that! Let me try this out then. – MathMan08 Apr 06 '13 at 23:04
  • I got $\pi$ for my answer. – MathMan08 Apr 06 '13 at 23:07
  • You are correct; I dropped a factor of $2$ in my answer. Good job. – Ron Gordon Apr 06 '13 at 23:10
  • You're welcome. Many times, cylindrical coordinates are a better call when you have a situation with axial symmetry like here. You can still use sphericals, but IMO they are more difficult. – Ron Gordon Apr 06 '13 at 23:12
  • Actually, CORRECTLY doing spherical here is simple. After the first integral you get the double integral of $\int_0^{2\pi}\int_0^{\pi/3}[4cos(\phi)sin(\phi)-2sin(\phi)]d\phi d\theta$. – MathMan08 Apr 06 '13 at 23:23
  • In lots of mvc questions here people put the $\rho(z^2+\rho^2)^{-3/2}$ after the $\mathrm d\rho$. Why is that? If it is not a constant, shouldn't it be before the differential? – John D Jul 26 '21 at 18:01