Definite Integral $\int_0^r\arcsin\left(\frac{\sqrt{r^2-x^2}}{R^2}\right)dx$

I am calculating the area of a circle (radius r) inscribed over a cylindrical surface (radius R).

How do I solve the following integral?

$$\int_0^r\arcsin\left(\frac{\sqrt{r^2-x^2}}{R^2}\right)dx$$

• Start with integration by parts. – Mhenni Benghorbal Jan 26 '14 at 7:05
• I have the bad feeling that this would lead to some elliptic integrals. – Claude Leibovici Jan 26 '14 at 7:16
• Something is wrong in your formula. Are you sure this is $R^2$ and not $R$ in the denominator ? A dimensional analysis shows also that your integral is a length, not an area. – Tom-Tom Feb 28 '14 at 9:23

Here is an approach. Recalling the power series of $\arcsin x$, we have

$$I = \sum _{k=0}^{\infty }{\frac { \left( 2\,k \right) !\,{4}^{-k}}{ \left( k! \right) ^{2} \left( 2\,k+1 \right) }} \int_{0}^{r}( r^2-x^2 )^{(2k+1)/2} \,dx .$$

Now, use the substitution $u=x/r$ and use the beta function to evaluate the integral, then resum the resulting series which will give you the answer in terms of Elliptic integrals

$$\frac{r^2}{R^2} K\left( {\frac {r}{{R}^{2}}} \right) -{R }^{2}{K} \left( {\frac {r}{{R}^{2}}} \right) +{R}^{2}{E} \left( {\frac {r}{{R}^{2}}} \right) .$$

You can use computer algebra systems to sum the resulting series.

Note: Double check my calculations.

• i am not from mathematical background. even never heard of 'Elliptic integrals'!!! seems i need to lot of study. thanks a lot to all. – user123910 Jan 26 '14 at 9:31
• @user123910: This is not an elementary integral, so do not expect to have a closed form in terms of ordinary functions. – Mhenni Benghorbal Jan 26 '14 at 20:57