I just need to know the bounds of integration for this:

Seawater has density 1025 kg/m and flows in a velocity field v=yi+xj. Find the rate of flow outward through the hemisphere x^2+y^2+z^2=9, z=0

Question

I just need to know the bounds of integration for this:

Seawater has density 1025 kg/m and flows in a velocity field v=yi+xj. Find the rate of flow outward through the hemisphere x^2+y^2+z^2=9, z=>0

anonymous · Answer

I converted the points to sphereical coordinates so I have phi and theta, but I'm not sure if the bounds are 0<theta<2pi or 0<theta<pi

anonymous · Answer

Since it says through the hemisphere and not the entire sphere wouldn't that make it 0<theta<pi not 2pi?

anonymous · Answer

@phi @jonnymiller I believe the formula for this type of problem is:
$$\rho \int\limits_{?}^{?}\int\limits_{?}^{?}(yi+xj)*(r _{\phi} \ X \ r _{\Theta})dA$$
I'm just not sure of the bounds of integration

phi · Answer

you want the upper half of a sphere (of radius 3) centered at (0,0,0)
if theta is the angle with the z axis, you want to go from 0 to pi/2
phi, the angle with the x-axis, would then be 0 to 2pi

You could figure this out by sketching the surface.  if you hold r=3 and set theta=pi/2 (i.e. horizontal) you would trace out a ring as you rotate 2pi around the z-axis.  Now change the phi angle a little bit and trace out another ring.
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