<s>[INST] Find the area above the $x-$ axis included between the parabola $y^2=ax$ and the circle $x^2+y^2=2ax$. 
Find the area above the $x-$ axis included between the parabola $y^2=ax$ and the circle $x^2+y^2=2ax$.

The circle $x^2+y^2=2ax$ can be represented as $(x-a)^2+y^2=a^2$ which has centre at $(a,0)$ and radius $a$.
Since they both intersect at $(0,0)$ and $(a,a)$ so the area included will be 
$\int _0^a \int _0^{\sqrt {ax}} dy dx-\int_0^a \int _0^{\sqrt {2ax-x^2}}dy dx$
But the problem is I am unable to compute the 2nd integral.
The answer to the first is $\frac{2}{3}a^{{2}}$.
Please help to solve it.
 [/INST] 
The graph is drawn taking $a=2$. What you need to notice is that the second integral is just $1/4$-th the area of the circle. So required area is $$\int_0^a\sqrt{ax}dx+\frac{\pi a^2}4=\frac{a^2}{3/2}+\frac{\pi a^2}4$$
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