The region trapped between y = = 4x - 2x² and 4x-2x² |y= ५-२x२-३ = 21² - 1³ y = a) Set up the integral that represents volume of the solid generated when the region is rotated about the I - axis. You do not have to integrate. b da a where a = and b =
The region trapped between y = = 4x - 2x² and 4x-2x² |y= ५-२x२-३ = 21² - 1³ y = a) Set up the integral that represents volume of the solid generated when the region is rotated about the I - axis. You do not have to integrate. b da a where a = and b =
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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![### Finding the Volume of the Solid of Revolution
This educational example demonstrates how to find the volume of a solid generated by rotating a region bounded by two curves around the x-axis.
In this example, the region is bounded by the curves:
\[ y = 4x - 2x^2 \]
and
\[ y = 2x^2 - x^3 \]
#### Graph Explanation
The graph shows the area between the two curves. The curve \( y = 4x - 2x^2 \) is labeled in red, while the curve \( y = 2x^2 - x^3 \) is labeled in blue. The shaded area represents the region between these two curves, which is to be rotated around the x-axis to generate a solid.
#### Setting Up the Integral
To find the volume of the solid generated by rotating the given region about the x-axis, we can use the method of disks or washers. The formula for the volume of the solid when rotating the specified region around the x-axis is given by:
\[ V = \pi \int_{a}^{b} \left[ f(x)^2 - g(x)^2 \right] dx \]
In this case:
- \( f(x) = 4x - 2x^2 \)
- \( g(x) = 2x^2 - x^3 \)
The limits of integration \( a \) and \( b \) are the points of intersection of the curves. Solving for these points will give the exact values.
Using the formula, the integral is set up as follows:
\[ \int_{a}^{b} \pi \left( (4x - 2x^2)^2 - (2x^2 - x^3)^2 \right) dx \]
The points \( a \) and \( b \) are the x-coordinates where the two functions intersect, and these can be found by solving:
\[ 4x - 2x^2 = 2x^2 - x^3 \]
#### Solving for Intersection Points
To determine the limits \( a \) and \( b \):
- Simplify and solve \( 4x - 2x^2 = 2x^2 - x^3 \)
Finally, fill in the values for \( a \) and \( b \) when they are found:
\[
\int_{a = \text{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F294bc7d3-33f6-4bdd-8888-0c517e9f6257%2F90526064-4dae-4fca-9a51-893b91314346%2Fe3oaox_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Volume of the Solid of Revolution
This educational example demonstrates how to find the volume of a solid generated by rotating a region bounded by two curves around the x-axis.
In this example, the region is bounded by the curves:
\[ y = 4x - 2x^2 \]
and
\[ y = 2x^2 - x^3 \]
#### Graph Explanation
The graph shows the area between the two curves. The curve \( y = 4x - 2x^2 \) is labeled in red, while the curve \( y = 2x^2 - x^3 \) is labeled in blue. The shaded area represents the region between these two curves, which is to be rotated around the x-axis to generate a solid.
#### Setting Up the Integral
To find the volume of the solid generated by rotating the given region about the x-axis, we can use the method of disks or washers. The formula for the volume of the solid when rotating the specified region around the x-axis is given by:
\[ V = \pi \int_{a}^{b} \left[ f(x)^2 - g(x)^2 \right] dx \]
In this case:
- \( f(x) = 4x - 2x^2 \)
- \( g(x) = 2x^2 - x^3 \)
The limits of integration \( a \) and \( b \) are the points of intersection of the curves. Solving for these points will give the exact values.
Using the formula, the integral is set up as follows:
\[ \int_{a}^{b} \pi \left( (4x - 2x^2)^2 - (2x^2 - x^3)^2 \right) dx \]
The points \( a \) and \( b \) are the x-coordinates where the two functions intersect, and these can be found by solving:
\[ 4x - 2x^2 = 2x^2 - x^3 \]
#### Solving for Intersection Points
To determine the limits \( a \) and \( b \):
- Simplify and solve \( 4x - 2x^2 = 2x^2 - x^3 \)
Finally, fill in the values for \( a \) and \( b \) when they are found:
\[
\int_{a = \text{
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