Question

R is the region : x^2+y^2<=1, x>=0,y<=0

Find the integral: ∫∫R e^(x^2+y^2)xydxdy

Answer 1

Change the variables: x=r.costheta, y=r.sintheta. Then the region would become 0<=r<=1, -pi/2<=theta<=0.
We now need to change the differentials. dxdy=dA=r.dr.dtheta.
You do the math.

Answer 2

ok i did that. But the integral is coming to ∫∫D e^(r^2)r^3cos(θ)sin(θ)drdθ.

Answer 3

Since the boundaries are constants, you can write this as \int(Dtheta) sintheta.costheta dtheta . \int(Dr) e^(r^2).r^3 dr. First part is easy.
For the second part, define s=r^2. Then integrate by parts with u=s, dv=e^s ds.