Monday Math 148

Find I=\int_{-1}^{1}\int_{\frac{x^2}2+x-\frac12}^{-\frac{x^2}2+x+\frac12}\frac{x^5-y^3}{x^2+y^2}\,dy\,dx.

One could integrate over y by using
\frac{x^5-y^3}{x^2+y^2}=\frac{x^5}{x^2+y^2}-\frac{y^3}{x^2+y^2}=\frac{x^5}{x^2+y^2}-y+\frac{x^2y}{x^2+y^2}; however, when you put in the limits, the resulting integral over x becomes very difficult.

Note that the region of integration, the region bounded by the parabolas y=\frac{x^2}2+x-\frac12 and y=-\frac{x^2}2+x+\frac12, has twofold rotational symmetry about the origin. Thus, we consider the change in variables u=-x,\;\;v=-y, which represents axes rotated by 180°. We then have
and if we rename u and v as x and y, respectively, we see
For our integrand f(x,y)=\frac{x^5-y^3}{x^2+y^2}, we have f(-x,-y)=-f(x,y), which combined with the rotational symmetry of the region of integration, means that the integral is zero.


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