Let R be the region defined by the graphs of: y = √(x-1), y = 0, and x = 5 Determine AREA of the region R by using integration with respect to x as well as by integrating with respect to y.
Let R be the region defined by the graphs of: y = √(x-1), y = 0, and x = 5 Determine AREA of the region R by using integration with respect to x as well as by integrating with respect to y.
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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![**Transcription and Explanation for Educational Context**
---
**Introduction:**
Let \( R \) be the region defined by the graphs of \( y = \sqrt{(x-1)} \), \( y = 0 \), and \( x = 5 \).
**Objective:**
Determine the area of the region \( R \) by using integration with respect to \( x \) as well as by integrating with respect to \( y \).
---
**Area Calculation**
1. **Area by Integrating with Respect to \( x \):**
\[
\text{Area} = \int_1^5 \sqrt{x-1} \, dx
\]
- Integration steps:
\[
= \left[ \frac{(x-1)^{3/2}}{\frac{3}{2}} \right]_1^5
\]
\[
= \left[ \frac{2}{3}(x-1)^{3/2} \right]_1^5
\]
\[
= \frac{2}{3} \left[ (4)^{3/2} - 0 \right]
\]
\[
= \frac{16}{3} \, \text{units}^2
\]
2. **Area by Integrating with Respect to \( y \):**
\[
\text{Area} = \int_0^2 (y^2 - 1) \, dy
\]
- Integration steps:
\[
= \left[ \frac{y^3}{3} - y \right]_0^2
\]
\[
= \left( \frac{8}{3} - 2 \right) - (0)
\]
\[
A = \frac{2}{3} \, \text{units}^2
\]
---
**Solid of Revolution:**
Now consider the above region \( R \) is revolved about the line \( x = 6 \).
1. **Determine Volume Using the Shell Method:**
- Formula and diagram not provided. Proceed with shell method calculations expected in this context.
2. **Determine Volume Using the Disk/Washer Method:**
\[
\int_c^d \left[ R(x)^2 - r(y)^](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e1fa912-5e0c-4d6e-bcb6-bc3942e62084%2F3a732be8-f5dd-4976-8331-a7bc9c0d384c%2Fo6hu3wl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription and Explanation for Educational Context**
---
**Introduction:**
Let \( R \) be the region defined by the graphs of \( y = \sqrt{(x-1)} \), \( y = 0 \), and \( x = 5 \).
**Objective:**
Determine the area of the region \( R \) by using integration with respect to \( x \) as well as by integrating with respect to \( y \).
---
**Area Calculation**
1. **Area by Integrating with Respect to \( x \):**
\[
\text{Area} = \int_1^5 \sqrt{x-1} \, dx
\]
- Integration steps:
\[
= \left[ \frac{(x-1)^{3/2}}{\frac{3}{2}} \right]_1^5
\]
\[
= \left[ \frac{2}{3}(x-1)^{3/2} \right]_1^5
\]
\[
= \frac{2}{3} \left[ (4)^{3/2} - 0 \right]
\]
\[
= \frac{16}{3} \, \text{units}^2
\]
2. **Area by Integrating with Respect to \( y \):**
\[
\text{Area} = \int_0^2 (y^2 - 1) \, dy
\]
- Integration steps:
\[
= \left[ \frac{y^3}{3} - y \right]_0^2
\]
\[
= \left( \frac{8}{3} - 2 \right) - (0)
\]
\[
A = \frac{2}{3} \, \text{units}^2
\]
---
**Solid of Revolution:**
Now consider the above region \( R \) is revolved about the line \( x = 6 \).
1. **Determine Volume Using the Shell Method:**
- Formula and diagram not provided. Proceed with shell method calculations expected in this context.
2. **Determine Volume Using the Disk/Washer Method:**
\[
\int_c^d \left[ R(x)^2 - r(y)^
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