Question

Decide, without calculation, if each of the integrals below are positive, negative, or zero. Let D be the region inside the unit circle centered at the origin. Let T, B, R, and L denote the regions enclosed by the top half, the bottom half, the right half, and the left half of unit circle, respectively.1. ∬B xe^xdA2. ∬R xe^xdA3. ∬T xe^xdA4. ∬D xe^xdA5. ∬L xe^xdA? Positive Negative Zero

Answer 1

The integrals over B and T will be positive. Keeping
`y` fixed,
`xe^x` is strictly increasing over D as
`x` increases, so the integrals over
`x<0` (i.e. the bottom/top left quadrants of D) is negative but the integrals over
`x>0` are *more* positive.

The integrals over R and L are zero. If we take
`f(x,y)=xe^x`, then
`f(x,-y)=f(x,y)`, which is to say
`f` is symmetric across the
`x`-axis. For the same reason, the integral over all of D is also zero.