How to solve the following $\iint\limits_{x^2+y^2\ge k}\frac{\exp(-(x^2+y^2)/2)}{2\pi}dxdy?$
I think I should make the substitution $u=x^2+y^2$, but I don't know how the integral will look like.
How to solve the following $\iint\limits_{x^2+y^2\ge k}\frac{\exp(-(x^2+y^2)/2)}{2\pi}dxdy?$
I think I should make the substitution $u=x^2+y^2$, but I don't know how the integral will look like.
If you use polar coordinates, you have $dx\,dy = r\,dr\,d\theta$, so that: $\iint\limits_{x^2+y^2\ge k}\frac{\exp(-(x^2+y^2)/2)}{2\pi}\,dx\,dy = \int\limits_{r\ge \sqrt{k}}\exp(-r^2/2)r\,dr$ You can then use $u = r^2/2,~ du = r \,dr$ to get: $\int\limits_{r\ge \sqrt{k}}\exp(-r^2/2)r\,dr = \int\limits_{u\ge k/2}\exp(-u)\,du = e^{-k/2}$