Let
D = {(x,y): x^2 + y^2 <= 1} the disk
B = {(x,y) in D, y <= 0} the bottom half of the disk
R = {(x,y) in D, x >= 0} the right half of the disk

Without integrating determine if the integrals are 
positive(+), negative(-) or zero(0)
a. double-integral over B of 5x dA
b. double-integral over R of (y^3 + y^5) dA
c. double-integral over D of cos(y) dA
d. double-integral over D of x e^x  dA

Answers:
a. zero  (x > 0, balances with x < 0)
b. zero  (odd function, y>0 balances with y<0)
c. positive (cos y > 0 for |y| <= 1)
d. positive (since e^x is bigger than e^(-x) for x>0, the negative
part of xe^x is smaller than the positive part)


All of these are really 1-variable problems, D is balanced in both directions
a. integral for -L to L of 5x dx           [B is balanced in x direction]
b. integral for -L to L of (y^3 + y^5) dy  [R is balanced in y direction]
c. integral for -L to L of cos(y) dy       
d. integral for -L to L of x e^x