Evaluate using polar coordinates: $\iint_Re^{-x^2-y^2}dA$, where $R$ is part of an annulus bounded by $y=|x|$

Question

$R$ is the portion of the annulus $\{(x,y):4\le\sqrt{x^2+y^2}\le9\}$ in the upper half plane, bounded by the graph $y=|x|$. Evaluate the integral $$\iint_Re^{-x^2-y^2}dA$$

My Try:

$$\sqrt{x^2+y^2}=4$$

$$x^2+y^2=16$$

$$\sqrt{x^2+y^2}=9$$

$$x^2+y^2=81$$

So, $4\le r\le9$ and $\dfrac{\pi}{4}\le\theta\le\dfrac{3\pi}{4}$ So, my integral will be

$$\int_{\pi/4}^{3\pi/4}\int_{4}^{9}re^{-r^2}dr\ d\theta$$

Is my above integral correct?

Answer 1

Yes, your integral is correct and you have the correct lower and upper bounds for $r$ and $\theta$