Integrating $\sqrt{r^2 - x^2 - z^2}$ I need to integrate the following piece:
$\int \sqrt{R^2 - x^2 - z^2} \, dx$. 
This is what I tried:


*

*$\int \sqrt{R^2 - x^2 - z^2} \, dx = \int (R^2 - x^2 - z^2)^\frac{1}{2} \, dx$

*Substitute $u = (R^2 - x^2 - z^2)^\frac{1}{2}, du = \frac{du}{dx}dx = \frac{1}{2}(R^2 - x^2 - z^2)^\frac{-1}{2} * 2x dx, dx = \frac{(R^2 - x^2 - z^2)^\frac{1}{2}}{x} du = \frac{u}{x}du$

*But I'm stuck.


The expected result is: $\frac{R^2- z^2}{2} * \arcsin(\frac{y}{\sqrt{R^2 - z^2}}) + \frac{y}{2}*\sqrt{R^2-z^2-y^2}$

Hint:
Let $R^2-z^2=a^2$. Then, substitute $x=a\sin\theta$.
$$I=\int\sqrt{a^2-x^2}dx=\int a^2\cos^2\theta d\theta$$
Use $\cos^2\theta=\frac{1+\cos2\theta}{2}$
