To evaluate the double integral $\int \int e^{x^2 + y^2} \, dy \, dx$ using polar coordinates, we will go through the following steps:

Step 1: Understand the expression and setup

We are given the integral:

$I = \int \int e^{x^2 + y^2} \, dy \, dx$

Step 2: Convert to polar coordinates

Recall that in polar coordinates, $x = r \cos \theta$ and $y = r \sin \theta$, and the element of area $dx \, dy$ transforms to $r \, dr \, d\theta$.

Thus: $x^2 + y^2 = r^2$ So the integrand $e^{x^2 + y^2}$ becomes $e^{r^2}$, and the area element becomes $r \, dr \, d\theta$.

Therefore, the double integral becomes:

$I = \int \int e^{r^2} \cdot r \, dr \, d\theta$

Step 3: Limits of integration

Since the limits of integration were not provided, let's assume that we are integrating over a circular region with radius $R$ centered at the origin. In polar coordinates, the limits for $r$ would range from 0 to $R$, and the limits for $\theta$ would range from 0 to $2\pi$.

Step 4: Set up the integral

Now, the double integral becomes:

$I = \int_0^{2\pi} \int_0^R e^{r^2} r \, dr \, d\theta$

Step 5: Perform the integration

We can first handle the $r$-integral:

$\int_0^R e^{r^2} r \, dr$

This is a standard integral, and we can make a substitution: Let $u = r^2$, so that $du = 2r \, dr$, which simplifies the integral. The limits for $u$ will be from 0 to $R^2$. Thus, the integral becomes:

$\frac{1}{2} \int_0^{R^2} e^u \, du = \frac{1}{2} \left[ e^u \right]_0^{R^2} = \frac{1}{2} \left( e^{R^2} - 1 \right)$

Now the integral becomes:

$I = \int_0^{2\pi} \frac{1}{2} \left( e^{R^2} - 1 \right) d\theta$

Step 6: Final integration

Now, integrate with respect to $\theta$:

$I = \frac{1}{2} \left( e^{R^2} - 1 \right) \int_0^{2\pi} d\theta$

The integral of $d\theta$ from 0 to $2\pi$ is simply $2\pi$, so the result is:

$I = \frac{1}{2} \left( e^{R^2} - 1 \right) \cdot 2\pi = \pi \left( e^{R^2} - 1 \right)$

Final Answer:

$I = \pi \left( e^{R^2} - 1 \right)$

Do you need further details or have any questions on this? Here are 5 related questions that could expand on this:

1. How do I compute the double integral of a function in polar coordinates with different region limits?
2. What is the significance of the Jacobian when transforming to polar coordinates in integrals?
3. How would the result change if we were integrating over a different region, say, a square or ellipse?
4. How do I handle integrals involving exponential functions of $r^2$ in other contexts?
5. What happens if we have an infinite region, such as $R \to \infty$?

Tip: When dealing with exponential integrals in polar coordinates, always check the form of the integrand and consider using substitution methods to simplify the expression.