For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis. 83. y = 2x³, x=0, x= 1 and y = 0
For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis. 83. y = 2x³, x=0, x= 1 and y = 0
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Exercise Instructions**
For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis.
**Exercise 83:**
\[ y = 2x^3, \, x = 0, \, x = 1, \, \text{and} \, y = 0 \]
---
**Detailed Explanation:**
1. **Identify the curves and boundaries:**
- The first equation \( y = 2x^3 \) represents a cubic function that increases steeply as \(x\) increases.
- The boundary \(x = 0\) represents the y-axis.
- The boundary \(x = 1\) is a vertical line at \(x = 1\).
- The boundary \(y = 0\) is the x-axis.
2. **Draw the region:**
- Plot the curve \( y = 2x^3 \). Start at the origin (0,0) and as \( x \) increases from 0 to 1, calculate some points to draw the curve. For instance, at \( x = 0.5 \), \( y = 2(0.5)^3 = 0.25 \).
- Draw the vertical line \( x = 0 \) (the y-axis).
- Draw the vertical line \( x = 1 \).
- Draw the horizontal line \( y = 0 \) (the x-axis).
- Shade the area bounded by these curves and lines.
3. **Rotate the region around the y-axis:**
- When rotating the shaded area around the y-axis, the shape will form a solid of revolution.
- To find the volume, use the method of disks or shells, as suitable.
4. **Calculate the volume:**
- You can use the washer method (since the region is bounded by the y-axis and another curve):
\[
V = \pi \int_{0}^{1} [1^2 - (1 - \text{function in terms} \, x)] dx
\]
- Substitute and simplify accordingly.
This procedure explains how to identify the bounded region, draw it, and calculate the volume when rotated around the y-axis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb35f0eab-2452-4d20-90d7-c62e398afc80%2F919961a0-c336-4540-816c-3da1a9a51426%2F6a2z775_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise Instructions**
For the following exercises, draw the region bounded by the curves. Then, find the volume when the region is rotated around the y-axis.
**Exercise 83:**
\[ y = 2x^3, \, x = 0, \, x = 1, \, \text{and} \, y = 0 \]
---
**Detailed Explanation:**
1. **Identify the curves and boundaries:**
- The first equation \( y = 2x^3 \) represents a cubic function that increases steeply as \(x\) increases.
- The boundary \(x = 0\) represents the y-axis.
- The boundary \(x = 1\) is a vertical line at \(x = 1\).
- The boundary \(y = 0\) is the x-axis.
2. **Draw the region:**
- Plot the curve \( y = 2x^3 \). Start at the origin (0,0) and as \( x \) increases from 0 to 1, calculate some points to draw the curve. For instance, at \( x = 0.5 \), \( y = 2(0.5)^3 = 0.25 \).
- Draw the vertical line \( x = 0 \) (the y-axis).
- Draw the vertical line \( x = 1 \).
- Draw the horizontal line \( y = 0 \) (the x-axis).
- Shade the area bounded by these curves and lines.
3. **Rotate the region around the y-axis:**
- When rotating the shaded area around the y-axis, the shape will form a solid of revolution.
- To find the volume, use the method of disks or shells, as suitable.
4. **Calculate the volume:**
- You can use the washer method (since the region is bounded by the y-axis and another curve):
\[
V = \pi \int_{0}^{1} [1^2 - (1 - \text{function in terms} \, x)] dx
\]
- Substitute and simplify accordingly.
This procedure explains how to identify the bounded region, draw it, and calculate the volume when rotated around the y-axis.
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