Not sure if this surface integral is right

[unscientific]

Homework Statement

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)

 

[haruspex]

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]

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]

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!