The image contains a mathematical problem in Persian, and I'll provide a full solution and explanation for the second question.

The second problem involves solving a double integral. The problem is:

$\int_{0}^{\infty} \int_{y}^{\infty} e^{-x^2} dx \, dy$

Solution:

1. Understanding the Integral:

   • The integral is a double integral where the integration first occurs with respect to $x$ from $y$ to $\infty$, and then with respect to $y$ from $0$ to $\infty$.

2. Changing the Order of Integration:

   • We can solve this by changing the order of integration. The given region can be described in terms of $x$ and $y$ as:
     • $0 \leq y \leq x$ and $0 \leq x < \infty$.
   • The double integral can be rewritten as: $\int_{0}^{\infty} \int_{0}^{x} e^{-x^2} dy \, dx$

3. Simplifying the Inner Integral:

   • Now, compute the inner integral with respect to $y$: $\int_{0}^{x} e^{-x^2} dy$ Since $e^{-x^2}$ is independent of $y$, the integral simplifies to: $e^{-x^2} \cdot y \Big|_{0}^{x} = e^{-x^2} \cdot (x - 0) = x e^{-x^2}$

4. Evaluating the Outer Integral:

   • Now, evaluate the outer integral: $\int_{0}^{\infty} x e^{-x^2} dx$ To solve this, use the substitution $u = x^2$, hence $du = 2x dx$, and the integral becomes: $\frac{1}{2} \int_{0}^{\infty} e^{-u} du$ The integral $\int_{0}^{\infty} e^{-u} du$ is a standard exponential integral and equals 1, so the final result is: $\frac{1}{2} \cdot 1 = \frac{1}{2}$

Final Answer:

The value of the double integral is $\frac{1}{2}$.

Would you like more details on any specific part of this solution or have any further questions?

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Related Questions:

1. What are the steps to changing the order of integration in a double integral?
2. How does one evaluate a double integral over a non-rectangular region?
3. What is the significance of the substitution method in evaluating integrals?
4. Can you explain how to recognize when a double integral is easier to solve by changing the order of integration?
5. What are the applications of double integrals in real-world problems?
6. How can we interpret the result of this integral geometrically?
7. What other methods could be used to solve this integral?
8. How do double integrals differ from iterated integrals?

Tip:

When solving double integrals, always consider the geometry of the region of integration first. Changing the order of integration can often simplify the problem significantly.