Spherical coordinate problem

1. The problem statement, all variables and given/known data

Evaluate the integral below, where H is the solid hemisphere x^2 + y^2 + z^2 ≤ 9, z ≤ 0

[tex]\iiint 8-x^2-y^2\,dx\,dy\,dz[/tex].

2. Relevant equations

none

3. The attempt at a solution

[tex]\int_{0}^{2\pi} \int_{\frac{\pi}{2}}^{\pi} \int_{0}^{3} (8-2p^2 \sin^2{\phi}) p^2 \sin{\phi}\ ,dp\,d\phi\, d\theta [/tex]

do I set up the integral right?
 

HallsofIvy

Science Advisor
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In polar coordinates [itex]x= \rho cos(\theta)sin(\phi)[/itex] and [itex]y= \rho sin(\theta)sin(\phi)[/itex] so
[tex] x^2+ y^2= \rho^2 cos^2(\theta) sin^2(\phi)+ \rho^2 sin^2(\theta) sin^2(\phi)[/tex]
[tex]= \rho^2 sin^2(\phi)(cos^2(\theta)+ sin^2(\theta))= \rho^2 sin^2(\phi)[/tex]
NOT "[itex]2\rho^2 sin^2(\phi)[/itex].
 
ye! sorry. 8-p^2*sin^2(phi)

but if I use [tex]= \rho^2 sin^2(\phi)[/tex] are the limit of this integral right for the question stated above.
 

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