#int_A x + y \ dA#
#= int_(-3)^(3) dx \ int_(-sqrt(9 -x^2))^(sqrt(9 -x^2)) dy qquad x + y#
# = int_(-3)^(3) dx qquad [xy + y^2/2]_( - sqrt(9 -x^2))^( sqrt(9 -x^2)) #
# = int_(-3)^(3) dx qquad 2xsqrt(9 -x^2) #
# = int_(-3)^(3) dx qquad d/(dx) [- 2/3(9 -x^2)^(3/2) ] #
# = [- 2/3(9 -x^2)^(3/2) ]_(-3)^(3) bb(= 0) #
From the symmetry, what you would expect
In polar:
#= int_0^(2 pi) d theta int_0^3 dr qquad r * ( r cos theta + r sin theta )#
#= int_0^(2 pi) d theta qquad [r^3/3 ( cos theta + sin theta )]_0^3#
#= 9 int_0^(2 pi) d theta qquad cos theta + sin theta #
#= 9 [ sin theta - cos theta ]_0^(2 pi)#
#= 9( [ 0 - 1 ] - [0 - 1] ) = 0#