Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![**Analyzing the Region Enclosed by Given Curves**
**Problem Statement:**
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \(x\) or \(y\). Draw a typical approximating rectangle.
**Given Curves:**
- \(y = \frac{2}{x}\)
- \(y = \frac{2}{x^2}\)
- \(x = 5\)
**Graphs and Descriptions:**
1. **Top Left Graph:**
- The graph shows the curves \(y = \frac{2}{x}\) and \(y = \frac{2}{x^2}\).
- The enclosed region is shaded in blue.
- An approximating rectangle is drawn in red, vertical in orientation, representing integration with respect to \(x\).
2. **Top Right Graph:**
- This graph illustrates the same curves with the region enclosed located in a different part of the coordinate system.
- The region is also shaded in blue.
- The approximating rectangle is horizontal, implying consideration for integration with respect to \(y\).
3. **Bottom Left Graph:**
- Similar curves are shown, with the enclosed region shaded in blue.
- An approximating rectangle is vertically aligned, suggesting integration with respect to \(x\).
4. **Bottom Right Graph:**
- Shows a clear section of the enclosed region shaded in blue.
- The rectangle is approximately horizontal, showing integration with respect to \(y\).
- This graph is marked with a correct symbol, indicating the correct approach for solving the area of the region.
**Conclusion:**
The area of the region can be found by integrating with respect to \(x\) or \(y\), as shown in the marked graph (bottom right). The correct integral setup leads to the area calculation:
**Area of the Region:**
\[
\frac{2}{5} \left( 5 \ln 5 - 2 \right) \quad (\text{Incorrect Answer, as per the provided X mark})
\]](https://content.bartleby.com/qna-images/question/8582e4ab-a8f8-4410-82c1-98e7a4b9cf23/2ff27cd8-0299-4a8d-a632-700f44ff4851/t5vswca_processed.png)
Transcribed Image Text:**Analyzing the Region Enclosed by Given Curves**
**Problem Statement:**
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \(x\) or \(y\). Draw a typical approximating rectangle.
**Given Curves:**
- \(y = \frac{2}{x}\)
- \(y = \frac{2}{x^2}\)
- \(x = 5\)
**Graphs and Descriptions:**
1. **Top Left Graph:**
- The graph shows the curves \(y = \frac{2}{x}\) and \(y = \frac{2}{x^2}\).
- The enclosed region is shaded in blue.
- An approximating rectangle is drawn in red, vertical in orientation, representing integration with respect to \(x\).
2. **Top Right Graph:**
- This graph illustrates the same curves with the region enclosed located in a different part of the coordinate system.
- The region is also shaded in blue.
- The approximating rectangle is horizontal, implying consideration for integration with respect to \(y\).
3. **Bottom Left Graph:**
- Similar curves are shown, with the enclosed region shaded in blue.
- An approximating rectangle is vertically aligned, suggesting integration with respect to \(x\).
4. **Bottom Right Graph:**
- Shows a clear section of the enclosed region shaded in blue.
- The rectangle is approximately horizontal, showing integration with respect to \(y\).
- This graph is marked with a correct symbol, indicating the correct approach for solving the area of the region.
**Conclusion:**
The area of the region can be found by integrating with respect to \(x\) or \(y\), as shown in the marked graph (bottom right). The correct integral setup leads to the area calculation:
**Area of the Region:**
\[
\frac{2}{5} \left( 5 \ln 5 - 2 \right) \quad (\text{Incorrect Answer, as per the provided X mark})
\]
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