问题444: 计算二重积分

积分区域如图所示

$$ \begin{aligned} & \iint_D\left|y-x^2\right| d \sigma \\ = & \iint_{D_1} y-x^2 d \sigma +\iint_{D_2} x^2-y d \sigma \\ = & \int_{-1}^1 d x \int_{x^2}^2\left(y-x^2\right) d y+\int_{-1}^1 d x \int_0^{x^2} x^2-y d y \\ = & \left.\int_{-1}^1\left(\frac{1}{2} y^2-x^2 y\right)\right|_{y=x^2} ^2 d x+\left.\int_{-1}^1\left(x^2 y-\frac{1}{2} y^2\right)\right|_{y=0} ^{x^2} d x \\ = & \int_{-1}^1\left(\frac{4}{2}-\frac{x^4}{2}-2 x^2+x^4\right)+\left(x^4-\frac{1}{2} x^4\right) d x \\ = & \int_{-1}^1 2-2 x^2+x^4 d x \\ = & \frac{46}{15} \end{aligned} $$

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