How would I perform this surface integral?

[Boltzman Oscillation]

Homework Statement

∫∫ F ⋅ ndτ over the spherical region x^2 + y^2 + z^2 = 25
given F = r^3 r i already converted the cartesian coordinates to spherical in F

Homework Equations

n = r[/B]

The Attempt at a Solution

I know I can plug in F into the equation and then dot it with r to get the following:

∫∫ r^3 dτ
r should be constant at r = 5 so i can take it out of the integral and substitute
5^3∫∫dτ
now how can i solve the ∫∫dτ? I would have to convert dτ into spherical coordinates? Note: the double integral should have a circle on it but I don't know how to add one here. Any help is useful.

 

[Dewgale]

$d\tau$ in this case is the area element for the surface of the sphere. It should be in spherical coordinates, yes. If you don't know what it is, you can calculate the volume element in spherical coordinates from the Jacobian, then make $r$ constant (thereby removing $dr$).

 

[Boltzman Oscillation]

$d\tau$ in this case is the area element for the surface of the sphere. It should be in spherical coordinates, yes. If you don't know what it is, you can calculate the volume element in spherical coordinates from the Jacobian, then make $r$ constant (thereby removing $dr$).

Ah so dτ = r^2sinΦdθdΦ, plugging in r = 5 and then into the equation I get:

5^5∫∫sinΦdθdΦ

solving the integrals i get
5^5(θ)cos(Φ)

 

[Dewgale]

Close. But if those are closed integrals, $\oint$ (pretend that's a double), it isn't an indefinite integral.

 

[haruspex]

how can i solve the ∫∫dτ?

Isn't it just the surface area of the sphere? Or do I misunderstand the notation?

 

[Boltzman Oscillation]

ah so i should consider my boundaries for a sphere? Where θ is from 0 to 2pi and φ is from 0 to pi.

answer should be 6250pi

 

[Dewgale]

You're off by a factor of 2, but yes, the method is correct.

 

[Boltzman Oscillation]

Got it, Thanks Dewgale.