Find the area of the region $\left\{\left(x,y\right) \bigg| \; 0 \leq y \le x^2 + 1, \; 0 \leq y \le x + 1, \; 0 \leq x \leq 2 \right\}$
The equation representing the inequation $0 \leq y \leq x^2 + 1$ is the parabola $y = x^2 + 1$
The equation representing the inequation $0 \leq y \leq x + 1$ is the line $y = x + 1$
The equation representing the inequation $0 \leq x \leq 2$ is the line $x = 2$
The required area is the shaded region OABCDO.
$\text{Area} \left(OABCDO\right) = \text{Area} \left(OABEO\right) + \text{Area} \left(EBCDE\right)$ $\;\;\; \cdots \; (1)$
Let $\text{Area} \left(OABEO\right) = \left|I_1\right|$
$\begin{aligned}
I_1 & = \displaystyle \int \limits_{0}^{1} y \; dx \text{ where } y = x^2 + 1 \\\\
& = \int \limits_{0}^{1} \left(x^2 + 1\right) \; dx \\\\
& = \left[\dfrac{x^3}{3} + x\right]_{0}^{1} = \dfrac{1}{3} + 1 = \dfrac{4}{3}
\end{aligned}$
$\therefore$ $\;$ $\text{Area} \left(OABEO\right) = \dfrac{4}{3}$ sq units $\;\;\; \cdots \; (2)$
Let $\text{Area} \left(EBCDE\right) = \left|I_2\right|$
$\begin{aligned}
I_2 & = \displaystyle \int \limits_{1}^{2} y \; dx \text{ where } y = x + 1 \\\\
& = \int \limits_{1}^{2} \left(x + 1\right) \; dx \\\\
& = \left[\dfrac{x^2}{2} + x\right]_{1}^{2} = \dfrac{4}{2} + 2 - \dfrac{1}{2} - 1 = \dfrac{5}{2}
\end{aligned}$
$\therefore$ $\;$ $\text{Area} \left(EBCDE\right) = \dfrac{5}{2}$ sq units $\;\;\; \cdots \; (3)$
$\therefore$ $\;$ In view of equations $(2)$ and $(3)$, equation $(1)$ becomes
$\text{Area} \left(OABCDO\right) = \dfrac{4}{3} + \dfrac{5}{2} = \dfrac{23}{6}$ sq units