Use Symmetry to Evaluate the Double Integral

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Use symmetry to evaluate the double integral. ∫∫R xy/1+x^4dA, R={(x,y) | -1 ≤x≤1, 0≤y≤1}

Answer: 0


First, notice that you can split the integral into the two single variable integrals.

use symmetry to evaluate the double integral.

The integral involving x may seem troublesome to integrate, but notice that this is an odd function over the symmetric domain with respect to 0. Hence the integral is equal to zero. Since the double integral can be written as a product of two integrals, the total integral is zero.

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