Not sure if this surface integral is right

unscientific · Feb 2, 2013

Homework Statement

Handwritten derivation of a volume integral: the divergence integral of A dot n over the volume equals the triple integral of (y^2 + y) dV, rewritten as iterated integrals with limits 0 to 1, then 0 to 1-x, then 0 to 1, ending with the result 1/4.

The Attempt at a Solution

I did the manual integration of part (i) and got an answer of 5/6 instead, I'm not sure which part is wrong.. For the surfaces, I start off with the surface in the x-z plane, then the slanted plane, then the y-z plane, then the top of the prism and finally the bottom of the prism.

For the slanted plane, I projected it onto the x-z plane to do the integration..

Part (a)

Handwritten surface integral setup: a sketch of a triangular prism with arrows on three axes, beside the flux integral of vector field A over five faces, each term written as an integral over S with its face's normal vector.

Handwritten surface integral working: the integral over S1 of x^2 ds, plus the integral over S2 of (xy^2 + x^2)/sqrt(2) ds, plus integrals over S3, S4 and S5, reduced to iterated integrals and summed to 1/3 + 1/3 + 1/6 = 5/6.

haruspex · Feb 2, 2013

I see two errors in the face by face method.
You have the sign of the normal wrong in the first and third faces. The direction is always outwards.
For the second face, you correctly had ∫xy²+x², but went wrong in the next line.

unscientific · Feb 3, 2013

haruspex said:

I see two errors in the face by face method.
You have the sign of the normal wrong in the first and third faces. The direction is always outwards.
For the second face, you correctly had ∫xy²+x², but went wrong in the next line.

Thanks! I will try again.

unscientific · Feb 3, 2013

haruspex said:

I see two errors in the face by face method.
You have the sign of the normal wrong in the first and third faces. The direction is always outwards.
For the second face, you correctly had ∫xy²+x², but went wrong in the next line.

Finally got the answer as 1/4 ! Thanks a lot!

Not sure if this surface integral is right

Homework Statement

The Attempt at a Solution

Discussion

Similar threads