Mathematics - Area Under Curves Question with Solution | TestHub

Given, equation of circle $x^2+y^2=2ax$ $\Rightarrow$ $(x-a{)}^2+y^2=a^2$ and equation of parabola is $y^2=ax,a>0$ Intersection points of circle and parabola $\begin{array}{lr}\Rightarrow & x^2+ax=2ax\\ \Rightarrow & x^2=ax\end{array}$ $\Rightarrow x^2-ax=0$ $\Rightarrow x(x-a)=0$ $\Rightarrow x=0,a$ Intersecting points are $(0,0)$ and $(a,a)$ $\therefore$ Required area $=\frac{\pi a^2}{4}-{\int }_0^a\sqrt{ax}dx$ $=\frac{\pi a^2}{4}-\sqrt{a}{\left(\frac{x^{3/2}}{3/2}\right)}_0^a$ $=\frac{\pi a^2}{4}-\frac{2a^2}{3}=a^2\left(\frac{\pi }{4}-\frac{2}{3}\right)$