Find .

One could integrate over *y* by using

; 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 and , has twofold rotational symmetry about the origin. Thus, we consider the change in variables , 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 , we have , which combined with the rotational symmetry of the region of integration, means that the integral is zero.

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Tags: Double Integral, Integral, Math, Monday Math, symmetry

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