https://file.aiquickdraw.com/mathbot/file/1726224199479dcm3uodk.jpg

The given double integral is:

\[
\int_0^1 \int_x^1 e^{-y^2} \, dy \, dx
\]

### Solution Outline:

1. **Change the order of integration**:
   - The integral is initially written with the inner integral in \(y\) and the outer integral in \(x\). We will change the order of integration to simplify the evaluation.

2. **Understand the region of integration**:
   - The limits are \(x\) from 0 to 1 and \(y\) from \(x\) to 1. This corresponds to the region where \(0 \leq x \leq 1\) and \(x \leq y \leq 1\).

3. **Change the limits**:
   - When swapping the order of integration, \(y\) will vary from 0 to 1, and for a fixed \(y\), \(x\) will vary from 0 to \(y\). The new integral becomes:
   
   \[
   \int_0^1 \int_0^y e^{-y^2} \, dx \, dy
   \]

4. **Evaluate the inner integral**:
   - The inner integral is:
   
   \[
   \int_0^y e^{-y^2} \, dx = e^{-y^2} \cdot y
   \]

5. **Evaluate the outer integral**:
   - Now, integrate with respect to \(y\):

   \[
   \int_0^1 y e^{-y^2} \, dy
   \]

   Let \( u = y^2 \), so \( du = 2y \, dy \), which transforms the integral to:

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

### Final Answer:

\[
\frac{1}{2} \left( 1 - e^{-1} \right)
\]

Would you like further clarifications or details on any part of this solution?

Here are 5 related questions to expand your understanding:
1. How does changing the order of integration simplify certain double integrals?
2. What is the geometric interpretation of the limits of integration?
3. How does the substitution method work in integrals, particularly with exponential functions?
4. What types of functions benefit most from changing the order of integration?
5. Can you apply the same technique of changing integration order to polar coordinates?

**Tip**: When changing the order of integration in double integrals, always carefully reconsider the region of integration to ensure the limits are properly transformed.

Evaluate the double integral: ∫₀¹ ∫ₓ¹ e^(-y²) dy dx

Solve the double integral ∫₀¹ ∫ₓ¹ e^(-y²) dy dx by changing the order of integration and using substitution. This advanced calculus problem demonstrates key concepts in multivariable integration.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation