How do you calculate the double integral of #(xcos(x+y))dr# where r is the region: 0 less than or equal to x less then or equal to #(2pi)/6#, 0 less than or equal to y less than or equal to #(2pi)/4#?
Could you clarify your question more? For one thing, you are sort of mixing rectangular and polar coordinates in the way you are stating it.
(I've reduced the fractions.)
We can choose whether to integrate first with respect to
I'll do the inner integral, then substitute.
Now we need to find
OK, maybe that wasn't easiest. Now I have to do 2 integration by parts, but rather than re-starting, we'll keep on this path.
We could replace
The second integral:
Using the same method, we'll get
Which evaluates to:
Adding the two integrals gives us: