How do I solve this double integral for a circular aperture with a y offset?

Question

I'm trying to solve the integral
$$intintfrac{dxdy}{sqrt{x^2+(y-y_0)^2+R^2}}$$
but not getting very far. It should have limits and in polar coordinates it is up to some fixed angle theta. R is a real constant and $y_0$ is an offset on the $y$ axis. If it's helpful it's the integral of a Lambertian source at some distance R, offset in y.

Narasimham · Answer

HINTS:
$$intintfrac{dx dy= dx; d(y-y_0)}{sqrt{x^2+(y-y_0)^2+R^2}}$$
Convert to polar coordinates
$$intintfrac{r dr d theta}{sqrt{x^2+(y-y_0)^2+R^2}}$$
$$ frac{theta_2-theta_1}{2} intfrac{d r^2}{sqrt{r^2+R^2}}$$
where due to offset reduced limit
$$theta_1= pi- sin^{-1}frac{y_0}{R};;theta_2= -theta_1 $$
Integral is in the form: $ ;2 sqrt{r^2+R^2} +c $

How do I solve this double integral for a circular aperture with a y offset?

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