Question

use symmetry to evaluate the double integral (1+x^2siny+y^2sinx)dA, R=[-pi,pi]x[-pi,pi].

let g(x) and h(y) be two functions:
int(c to d)int(a to b)(g(x,y)+h(x,y))dxdy=int(c to d)int(a to b)g(x,y)dxdy+int(c to d)int(a to b)h(x,y)dxdy

Answer 1

∫[-π,π]∫[-π,π] 1+x^2 siny + y^2 sinx dy dx
= ∫[-π,π] 2π dx
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