Example 3. Change the order of integration in the double integral $I=\int_{0}^{1} d x \int_{\sqrt{x}}^{2-x} f(x, y) d y$. Sketch the region of integration.
Solution. First, we need to construct the region of integration $D$. The integration in the inner integral is performed over $y$ from the parabola $y=\sqrt{x}$ to the line $y=2-x$, and in the outer integral - over $x$ from the point $x=0$ to the point $x=1$ (Fig. 3.6).

If we change the order of integration, then the inner integration will be performed over $x$ from the left boundary of the region $D$ (from the $O y: x=0$) to the right boundary - the curve $O A B$. However, this curve is described by different equations on different segments:

$$
x=\left[\begin{array}{r}
y^{2} \text { for } 0 \leqslant y \leqslant 1 \\
2-y \text { for } 1<y \leqslant 2
\end{array}\right.
$$

Therefore, the result will be written not as a single double integral, but as the sum of two integrals. Indeed, at the point $A$ with ordinate $y=1$, the equation describing the right boundary of the region $D$ changes, so the line $y=1$ divides the region $D$ into two subregions $D_{1}$ and $D_{2}$, each of which has a right boundary described by only one equation, and the integral over $D$ is represented as the sum of integrals over $D_{1}$ and $D_{2}$.

$$
\text { Answer. } I=\int_{0}^{1} d y \int_{0}^{y^{2}} f(x, y) d x+\int_{1}^{2} d y \int_{0}^{2-y} f(x, y) d x
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_446938f1e4808e7d6e07g-051.jpg?height=451&width=477&top_left_y=1120&top_left_x=1407)

Fig. 3.6

![](https://cdn.mathpix.com/cropped/2024_05_22_446938f1e4808e7d6e07g-051.jpg?height=437&width=447&top_left_y=1130&top_left_x=1977)

Fig. 3.7