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

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tikekeguer9305 tikekeguer9305

04-11-2019
Mathematics

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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

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LammettHash LammettHash · Answer 1 · 2019-11-05T15:49:36+01:00

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

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