Question

(1 point) 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

Answer 1

1. positive

2. positive

3. positive

4. positive

5. negative

Reasoning:

$xe^x$ is positive for positive $x$, and negative for negative $x$, but the exponential factor makes the positive part dominating, so that the integral over D is positive overall.

The integrals over B and T are identical by symmetry, and both are equal to half the value of the integral over D, so they are both positive as well.

The integral over L is negative because it takes strictly non-positive values of $x$.

Similarly, the integral over R is positive because it considers non-negative values of $x$.