eDoubleIntegral

Example Find the following double integral.

$\displaystyle \iint_{\Omega}(x^2 - y)dxdy, \ \Omega = \{-1 \leq x \leq 1, -x^2 \leq y \leq x^2\}$

Answer Draw a region $\Omega$ , we have

**図 1:** $-x^2 \leq y \leq x^2$
$\includegraphics[width=6cm]{CALCFIG/Fig7-2-2-4.eps}$

This region is V-simple. So, we can integrate in the direction of $y$ -axis first. Then we integrate in the direction of $x$ -axis.

$\displaystyle \iint_{\Omega}(x^2 - y)dxdy$	$\displaystyle =$	$\displaystyle \int_{-1}^{1}(\int_{-x^2}^{x^2}(x^2 - y)dy)dx$
	$\displaystyle =$	$\displaystyle \int_{-1}^{1}\left[(x^2 y - \frac{1}{2}y^2) \right ]_{-x^2}^{x^2} dx$
	$\displaystyle =$	$\displaystyle \int_{-1}^{1}[(x^4 - \frac{1}{2}x^4) - (-x^4 - \frac{1}{2}x^4)]dx$
	$\displaystyle =$	$\displaystyle \int_{-1}^{1}2x^4dx = \left[\frac{2}{5}x^5]\right ]_{-1}^{1} = \frac{4}{5}$