Multi variable integral : $\int_{0}^{1}\int_{0}^{\sqrt{1-x^2}} \frac{e^y}{\sqrt{1-x^2-y^2}} dy dx$

Question

How can I solve the following multi variable integral:

$\int_{0}^{1}\int_{0}^{\sqrt{1-x^2}} \frac{e^y}{\sqrt{1-x^2-y^2}} dy dx$

I have seen it in an exam. I tried to rewrite it in polar coordinates as follows, but to no avail.

$x^2+y^2 = r ^2$

$y = r.sin(\theta)$

$x = r.cos(\theta)$

$dydx = r dr d\theta$

$\int_{0}^{\pi/2}\int_{0}^{1} \frac{e^{r.sin\theta}}{\sqrt{1-r^2}} r dr d\theta$

If you are going to integrate in the stated order of $dxdy$ then you need to change the upper $x$ limit ot $\sqrt{1-y^2}$. — John Wayland Bales, Oct 03 '23 at 19:35

score 3 · Accepted Answer · answered Oct 03 '23 at 20:22

Simply swap the order of integration

$$I = \int_0^1\int_0^{\sqrt{1-y^2}}\frac{e^y}{\sqrt{1-y^2-x^2}}dxdy = \int_0^1\left[e^y\sin^{-1}\left(\frac{x}{\sqrt{1-y^2}}\right)\right]_0^{\sqrt{1-y^2}}dy$$

$$=\int_0^1\frac{\pi}{2}e^ydy= \boxed{\frac{\pi(e-1)}{2}}$$

Multi variable integral : $\int_{0}^{1}\int_{0}^{\sqrt{1-x^2}} \frac{e^y}{\sqrt{1-x^2-y^2}} dy dx$

1 Answers1