First Integral: Double Integral of $e^{-(x^2 + y^2)}$

We are asked to evaluate the double integral of the function $e^{-(x^2 + y^2)}$ from 0 to infinity in both the $x$- and $y$-directions:

$I = \int_0^\infty \int_0^\infty e^{-(x^2 + y^2)} \, dx \, dy$

To convert this double integral into polar coordinates, we follow these steps:

1. Convert to Polar Coordinates:

   In polar coordinates, the relationships are:

   • $x = r \cos \theta$
   • $y = r \sin \theta$
   • The differential area element becomes $dx \, dy = r \, dr \, d\theta$.

   The function $e^{-(x^2 + y^2)}$ becomes: $e^{-(x^2 + y^2)} = e^{-r^2}$ where $r^2 = x^2 + y^2$.

2. Adjust the Limits:

   Since the limits of integration are from 0 to infinity for both $x$ and $y$, the corresponding polar coordinate limits for $r$ go from 0 to infinity, and $\theta$ ranges from 0 to $\frac{\pi}{2}$ (as we are working within the first quadrant).

3. Rewrite the Integral in Polar Coordinates:

   The integral becomes: $I = \int_0^{\frac{\pi}{2}} \int_0^\infty e^{-r^2} r \, dr \, d\theta$

4. Evaluate the Inner Integral:

   We can first evaluate the $r$-integral: $\int_0^\infty e^{-r^2} r \, dr$ This is a standard Gaussian integral, which is known to evaluate to: $\frac{1}{2}$

5. Evaluate the Outer Integral:

   The outer integral with respect to $\theta$ is straightforward: $\int_0^{\frac{\pi}{2}} d\theta = \frac{\pi}{2}$

6. Combine Results:

   The final result of the double integral is: $I = \frac{1}{2} \times \frac{\pi}{2} = \frac{\pi}{4}$

Second Integral: Single Integral of $e^{-t^2}$

We are asked to evaluate the following integral: $\int_0^\infty e^{-t^2} \, dt$

This is also a standard Gaussian integral. The value of the Gaussian integral is well-known: $\int_{-\infty}^{\infty} e^{-t^2} \, dt = \sqrt{\pi}$

However, since the limits are from 0 to infinity (half of the total range), we divide the result by 2: $\int_0^\infty e^{-t^2} \, dt = \frac{\sqrt{\pi}}{2}$

Summary:

• The value of the double integral is $\frac{\pi}{4}$.
• The value of the single integral is $\frac{\sqrt{\pi}}{2}$.

Would you like further details on any of these steps or have additional questions?

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Further Questions:

1. How can you convert more complex integrals into polar coordinates?
2. What is the relationship between Gaussian integrals and their applications in probability theory?
3. How does the result of the single integral relate to the error function (erf)?
4. Can you extend these integrals to higher dimensions?
5. How does the factor $r$ arise in polar coordinate transformations of integrals?

Tip: When working with Gaussian integrals, recognize that they often appear in problems involving heat diffusion, probability distributions (normal distribution), and quantum mechanics.