https://file.aiquickdraw.com/mathbot/file/1726383219542do1wbnig.jpg

The image shows a double integral problem. The integral is as follows:

\[
(b) \int_1^{\ln 8} \int_0^{\ln y} e^{x + y} \, dx \, dy
\]

This is a double integral with respect to \( x \) and \( y \), where the limits for \( x \) range from \( 0 \) to \( \ln y \), and the limits for \( y \) range from \( 1 \) to \( \ln 8 \). The integrand is \( e^{x + y} \).

### Steps to Solve:

1. **Integrate with respect to \( x \):**
   The first step is to compute the inner integral:
   \[
   \int_0^{\ln y} e^{x + y} \, dx
   \]
   Since \( e^{x + y} = e^x \cdot e^y \), we can factor out \( e^y \):
   \[
   e^y \int_0^{\ln y} e^x \, dx
   \]
   The integral of \( e^x \) with respect to \( x \) is simply \( e^x \). Evaluating this from \( 0 \) to \( \ln y \):
   \[
   e^y \left[ e^{\ln y} - e^0 \right] = e^y \left( y - 1 \right)
   \]

2. **Integrate with respect to \( y \):**
   Now we integrate the result with respect to \( y \):
   \[
   \int_1^{\ln 8} e^y (y - 1) \, dy
   \]
   This can be broken into two integrals:
   \[
   \int_1^{\ln 8} y e^y \, dy - \int_1^{\ln 8} e^y \, dy
   \]
   
   The second term is straightforward:
   \[
   \int_1^{\ln 8} e^y \, dy = e^y \bigg|_1^{\ln 8} = e^{\ln 8} - e^1 = 8 - e
   \]

   For the first term, we use integration by parts. Let:
   \[
   u = y, \quad dv = e^y \, dy \quad \Rightarrow \quad du = dy, \quad v = e^y
   \]
   Applying integration by parts:
   \[
   \int y e^y \, dy = y e^y - \int e^y \, dy = y e^y - e^y
   \]
   Evaluating this from \( 1 \) to \( \ln 8 \):
   \[
   \left( y e^y - e^y \right) \bigg|_1^{\ln 8} = \left( \ln 8 \cdot 8 - 8 \right) - \left( 1 \cdot e - e \right) = 8 \ln 8 - 8 - e + e = 8 \ln 8 - 8
   \]

3. **Final Answer:**
   Putting it all together:
   \[
   (8 \ln 8 - 8) - (8 - e) = 8 \ln 8 - 16 + e
   \]

Thus, the solution to the integral is:

\[
8 \ln 8 - 16 + e
\]

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How does integration by parts work and when should it be applied?
2. Why can \( e^{x+y} \) be separated as \( e^x \cdot e^y \) in the integrand?
3. What are some strategies to approach complex limits in double integrals?
4. How would you approach this integral if the limits were different, say, from 0 to \( y \) instead of \( \ln y \)?
5. What happens if you reverse the order of integration for this double integral?

**Tip:** When dealing with exponential integrals, factoring and separating terms can simplify the calculation significantly.

Evaluate the double integral \( \int_1^{\ln 8} \int_0^{\ln y} e^{x + y} dx \, dy \)

Learn how to solve the double integral \( \int_1^{\ln 8} \int_0^{\ln y} e^{x + y} dx \, dy \) involving exponential functions. This solution covers step-by-step integration with respect to both variables, factoring the exponential function, and applying integration by parts. Suitable for university-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation