The problem requires evaluating the given iterated integrals by converting them to polar coordinates. Let’s analyze both integrals step by step.

1. Problem (1):

$\int_{0}^{2} \int_{0}^{\sqrt{4-x^2}} e^{-(x^2 + y^2)} \, dy \, dx$

Step 1: Conversion to polar coordinates

In polar coordinates:

• $x = r\cos\theta$, $y = r\sin\theta$
• $x^2 + y^2 = r^2$
• The Jacobian of the transformation is $dx\,dy = r \, dr\,d\theta$.

The limits:

• The region corresponds to $x^2 + y^2 \leq 4$ in the first quadrant ($0 \leq x \leq 2, 0 \leq y \leq \sqrt{4 - x^2}$).
• In polar coordinates, this region becomes $0 \leq r \leq 2$ and $0 \leq \theta \leq \frac{\pi}{2}$.

The integral becomes: $\int_{0}^{\frac{\pi}{2}} \int_{0}^{2} e^{-r^2} \, r \, dr \, d\theta$

Step 2: Solving the integral

Separate the integral: $\int_{0}^{\frac{\pi}{2}} d\theta \cdot \int_{0}^{2} r e^{-r^2} \, dr$

• The $\theta$-integral is: $\int_{0}^{\frac{\pi}{2}} d\theta = \frac{\pi}{2}$

• For the $r$-integral, use substitution $u = r^2$, (du = 2r ,