How can i solve this double integral $$ \iint_D x(y+x^2)e^{y^2-x^4} dxdy $$ where $$D=\{(x,y) \in \mathbb{R}^2: x^2 \leq y \leq x^2+1, 2-x^2 \leq y \leq 3-x^2 \}? $$
How to solve this double integral with a not-normal domain
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integration
multivariable-calculus
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0Try changing the variables, using $u=x^2$ and $y$ as the new variables. – 2012-02-26
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0I can write $D=\{(x,y)\in \mathbb{R}^2: 0 \leq y - x^2 \leq 1, 2 \leq y + x^2 \leq 3 \}$ and the integrand funcion as $f(x,y) =x(y+x^2)e^{(y-x^2)(y+x^2)}$. Sorry i wrote the wrong function first – 2012-02-26
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0With this notation i think that a possible change of variables is $u = y-x^2$ and $v=y+x^2$ but i don't know how to proceding – 2012-02-26
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0It looks to me as if because of the antisymmetry there is nothing to do, the integral is $0$. – 2012-02-26
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0Excuse me but i can't understand the "antisymmetry" of what? – 2012-02-26
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2@Katy23: The region of integration is symmetric about the $y$-axis. The function you are integrating has (beside the $x^2$ stuff) an $x$ in front. So the function you are integrating is *odd* $(f(-x)=-f(x)$. The integral over the region to the left of the $y$ axis cancels the integral on the right. It is basically the same fact as $\int_{-a}^a x\cos(x^2)=0$. – 2012-02-26
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0Perfect! You are right – 2012-02-26