Volume Integral of Cone

[geft]

I have the cone x^2 + y^2 <= z^2 with |z| <= 2
The vector function F = (4x, 3z, 5y)

With the divergence theorem I managed to reduce the equation to
∫∫∫ 4 dxdydz

Now the problem is finding out the limits. I know z goes from 0 to 2, but what about x and y?

 

[LCKurtz]

I have the cone x^2 + y^2 <= z^2 with |z| <= 2
The vector function F = (4x, 3z, 5y)

With the divergence theorem I managed to reduce the equation to
∫∫∫ 4 dxdydz

Now the problem is finding out the limits. I know z goes from 0 to 2, but what about x and y?

You haven't stated the problem. Are you calculating a flux integral? You might try writing your volume integral in cylindrical coordinates.

 

[geft]

The question asks to evaluate ∫F.ndA by the divergence theorem. I can just take a shortcut and use the general formula for cone volume, but I was wondering if there are known limits for the integrals.

 

[LCKurtz]

The question asks to evaluate ∫F.ndA by the divergence theorem. I can just take a shortcut and use the general formula for cone volume, but I was wondering if there are known limits for the integrals.

Of course there are. Integrate z first from z on the cone to z on the top and use polar coordinates for the dxdy integral. That is why I suggested cylindrical coordinates.

 

[SteamKing]

Your integral is very simple and can be evaluated without explicitly determining the limits of each integral. (Hint: what is the integral of dV of a cone over the entire volume of the cone?)

 

[LCKurtz]

The question asks to evaluate ∫F.ndA by the divergence theorem. I can just take a shortcut and use the general formula for cone volume, but I was wondering if there are known limits for the integrals.

Your integral is very simple and can be evaluated without explicitly determining the limits of each integral. (Hint: what is the integral of dV of a cone over the entire volume of the cone?)

Apparently he already knows that.