Integral of $\sqrt{ax-x^2}$ Can we integrate a function of the form $$\sqrt{ax-x^2},\text{say for any limits a to b}$$
I was solving a question recently where I had to find the volume and the eqn of base was $$x^2+y^2=ax\implies\,(x-\frac{a}{2})^2+y^2=\frac{a^2}{4}$$
now although it is easy to integrate this when writing x in terms of y, i tried to write y in terms of x and got the equation written above and was stuck as to how to integrate that form

Yes. Try the following:
\begin{align*}
ax-x^{2} &=-(x^{2}-ax) \\
&= \frac{a^{2}}{4}-(x^{2}-ax+\frac{a^{2}}{4})\\
&= \frac{a^2}{4}-\left(x-\frac{a}{2}\right)^{2}
\end{align*}
Then try substituting $x-\frac{a}{2}=\frac{a}{2}\sin(\theta)$.
