Hello, rmpatel5!
Who asked this question ??
It has nothing to do with a circle . . .
$\displaystyle \int^{x_1}_{x_o}(ax + b)\,dx$
Is this suffice to calculate the area of a circle segment? . . . . certainly not! Code:

* * *
*  *
*  :*
*  ::*
 :::
*  :::*
  *     +  a+:::*r 
*  :::*
 :::
*  ::*
*  :*
*  *
* * *

We have: .$\displaystyle x^2+y^2 \:=\:r^2 \quad\Rightarrow\quad y \:=\:\sqrt{r^2  x^2}$ . (the upper semicircle)
The area is: .$\displaystyle A \;=\;2 \times \int^r_a \left(r^2  x^2\right)^{\frac{1}{2}}\,dx $