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

Mathematics Asked on December 13, 2021

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.

One 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 $

Answered by Narasimham on December 13, 2021

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