4) ### Volume of Solids of Revolution - Calculus Problem#### Problem Statement:Find the volumes of the solids obtained by rotating the region bounded by the curves $ y = \frac{4}{\sqrt{x}} $ and $ y = x^2 $ about the following lines. Draw the graph in each case. **All answers must be in simplified form.**#### Subproblems:a.) The x-axis. **Work needs to be justified.**b.) The y-axis. **Work needs to be justified.**c.) $ y = 2 $. **Set up but do not evaluate.**#### Explanation:1. **Rotating about the x-axis:** - Find the points of intersection of the curves $ y = \frac{4}{\sqrt{x}} $ and $ y = x^2 $. - Use the disk method or washer method to set up the integral that represents the volume. - Integrate and simplify the result.2. **Rotating about the y-axis:** - Find the points of intersection of the curves. - Use the shell method to set up the integral that represents the volume. - Integrate and simplify the result.3. **Rotating about the line $ y = 2 $:** - Find the points of intersection of the curves. - Use the washer method, adjusting the radii to account for the offset of the line $ y = 2 $. - Set up the integral without performing the evaluation.#### Visualization:- **Graphs:** - Draw the graphs of both equations $ y = \frac{4}{\sqrt{x}} $ and $ y = x^2 $ to identify the region bounded by them. - Clearly mark the axes and the lines of rotation (x-axis, y-axis, and $ y = 2 $) on the graphs. - Shade the region that will be revolved to form the solid.This structured approach breaks down the problem into manageable steps and uses visualization to aid understanding, providing a clear path to solving these types of calculus problems.

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Description: 4) ### Volume of Solids of Revolution - Calculus Problem#### Problem Statement:Find the volumes of the solids obtained by rotating the region bounded by the curves $ y = \frac{4}{\sqrt{x}} $ and $ y = x^2 $ about the following lines. Draw the gra

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Find the volumes of the solids obtained by rotating the region bounded by the curves y=√√x and y=x² about the following lines. Draw the graph in each case. All answers must be in simplified form. a.) The x-axis. Work needs to be justified! (32 b.) The y-axis. Work needs to be justified. C c.) y 2. (Set up but do not evaluate). Gri

Find the volumes of the solids obtained by rotating the region bounded by the curves y=√√x and y=x² about the following lines. Draw the graph in each case. All answers must be in simplified form. a.) The x-axis. Work needs to be justified! (32 b.) The y-axis. Work needs to be justified. C c.) y 2. (Set up but do not evaluate). Gri

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

### Volume of Solids of Revolution - Calculus Problem

#### Problem Statement:
Find the volumes of the solids obtained by rotating the region bounded by the curves $ y = \frac{4}{\sqrt{x}} $ and $ y = x^2 $ about the following lines. Draw the graph in each case. **All answers must be in simplified form.**

#### Subproblems:
a.) The x-axis. **Work needs to be justified.**

b.) The y-axis. **Work needs to be justified.**

c.) $ y = 2 $. **Set up but do not evaluate.**

#### Explanation:
1. **Rotating about the x-axis:**
- Find the points of intersection of the curves $ y = \frac{4}{\sqrt{x}} $ and $ y = x^2 $.
- Use the disk method or washer method to set up the integral that represents the volume.
- Integrate and simplify the result.

2. **Rotating about the y-axis:**
- Find the points of intersection of the curves.
- Use the shell method to set up the integral that represents the volume.
- Integrate and simplify the result.

3. **Rotating about the line $ y = 2 $:**
- Find the points of intersection of the curves.
- Use the washer method, adjusting the radii to account for the offset of the line $ y = 2 $.
- Set up the integral without performing the evaluation.

#### Visualization:
- **Graphs:**
- Draw the graphs of both equations $ y = \frac{4}{\sqrt{x}} $ and $ y = x^2 $ to identify the region bounded by them.
- Clearly mark the axes and the lines of rotation (x-axis, y-axis, and $ y = 2 $) on the graphs.
- Shade the region that will be revolved to form the solid.

This structured approach breaks down the problem into manageable steps and uses visualization to aid understanding, providing a clear path to solving these types of calculus problems.

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