integral SS₁₁) dx dy xy=1, y = x, x=2 Calculate the in D limiteel to
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Calculate the Integral**
\[
\iint_D \left(\frac{x}{y}\right)^2 dx\,dy
\]
in \( D \) limited to \( xy = 1 \), \( y = x \), \( x = 2 \)
**Steps to Solve:**
1. **Identify the Region \( D \)**:
- The curve \( xy = 1 \): This is a hyperbola.
- The line \( y = x \): This is a diagonal line passing through the origin.
- The vertical line \( x = 2 \): This is a straight vertical line.
2. **Graphical Representation**:
- Sketch the hyperbola, focusing on the branch where \( x \) and \( y \) are positive.
- Draw the line \( y = x \), noting where it intersects the branch of the hyperbola.
- Add the vertical line \( x = 2 \) to the graph.
3. **Determine the Bounds**:
- The intersection of \( xy = 1 \) and \( y = x \) gives the point \( (1, 1) \).
- For \( x \), the bounds are from 1 to 2.
- For \( y \), the bounds are determined by the hyperbola equation and line equation within the given \( x \) range.
4. **Setup the Integral**:
Using the bounds:
\[
\iint_D \left(\frac{x}{y}\right)^2 dx\,dy
\]
Translate it into the iterated form:
\[
\int_{1}^{2} \int_{1}^{x} \left(\frac{x}{y}\right)^2 dy\, dx
\]
5. **Solve the Integral**:
Evaluate the inner integral first, followed by the outer integral.
This explanation provides a comprehensive method to set up and calculate the given double integral within the specified region.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F08f66248-6264-4fc4-bde2-ba87505c4c1b%2F566425e3-0c83-4607-a905-905d828c054b%2Fjdkoyif_processed.png&w=3840&q=75)
Transcribed Image Text:**Calculate the Integral**
\[
\iint_D \left(\frac{x}{y}\right)^2 dx\,dy
\]
in \( D \) limited to \( xy = 1 \), \( y = x \), \( x = 2 \)
**Steps to Solve:**
1. **Identify the Region \( D \)**:
- The curve \( xy = 1 \): This is a hyperbola.
- The line \( y = x \): This is a diagonal line passing through the origin.
- The vertical line \( x = 2 \): This is a straight vertical line.
2. **Graphical Representation**:
- Sketch the hyperbola, focusing on the branch where \( x \) and \( y \) are positive.
- Draw the line \( y = x \), noting where it intersects the branch of the hyperbola.
- Add the vertical line \( x = 2 \) to the graph.
3. **Determine the Bounds**:
- The intersection of \( xy = 1 \) and \( y = x \) gives the point \( (1, 1) \).
- For \( x \), the bounds are from 1 to 2.
- For \( y \), the bounds are determined by the hyperbola equation and line equation within the given \( x \) range.
4. **Setup the Integral**:
Using the bounds:
\[
\iint_D \left(\frac{x}{y}\right)^2 dx\,dy
\]
Translate it into the iterated form:
\[
\int_{1}^{2} \int_{1}^{x} \left(\frac{x}{y}\right)^2 dy\, dx
\]
5. **Solve the Integral**:
Evaluate the inner integral first, followed by the outer integral.
This explanation provides a comprehensive method to set up and calculate the given double integral within the specified region.
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