$\text{ The area of the region }\left\{(x,y):0\le y\le x^2+1,0\le y\le x+1,0\le x\le 2\right\}\text{ is }$

Equations of given curves are

$y=x^2+1\dots \dots \dots \dots \dots \dots$(1)

$y=x+1\dots \dots \dots \dots \dots \dots \dots .\left(2\right)$

 Solving (1) & (2)

$x^2+1=x+1\Rightarrow x^2-x=0$

$x=0,x=1$

$y=1,y=2$

$\text{ (1) }\underset{¯}{\underset{¯}{\&}\left(2\right)}\text{ intersect at }A(0,1)\text{ and }B(1,2)$

$RA=\underset{0}{\overset{1}{\int }}\left(x^2+1\right)dx+\underset{1}{\overset{2}{\int }}\left(x+1\right)dx$

$={\left(\frac{x^3}{3}+x\right)}_0^1+{\left(\frac{x^2}{2}+x\right)}_1^2=\frac{23}{6}=3.83$