Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = x² + 8 y = -x² + 2x + 12 x = 0 X = 3 Step 1 To find the point(s) of intersection of the curves y = x² + 8 and y = -x² + 2x + 12, equate both equations and solve. (x - x² - 2x - x² - x - )(x + x² + 8 = -x² + 2x + 12 = 0 = 0 ) = 0 X = , X =
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = x² + 8 y = -x² + 2x + 12 x = 0 X = 3 Step 1 To find the point(s) of intersection of the curves y = x² + 8 and y = -x² + 2x + 12, equate both equations and solve. (x - x² - 2x - x² - x - )(x + x² + 8 = -x² + 2x + 12 = 0 = 0 ) = 0 X = , X =
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 a Solid Generated by Revolving a Region Around the x-axis
To find the volume of the solid generated by revolving the region bounded by the graphs of the equations around the x-axis, follow these steps:
Given equations:
\[ y = x^2 + 8 \]
\[ y = -x^2 + 2x + 12 \]
\[ x = 0 \]
\[ x = 3 \]
#### Step 1:
**To find the point(s) of intersection of the curves \( y = x^2 + 8 \) and \( y = -x^2 + 2x + 12 \).**
We equate the two equations and solve for x.
\[ x^2 + 8 = -x^2 + 2x + 12 \]
Rearrange all terms to one side of the equation for standard quadratic form:
\[ x^2 + 8 + x^2 - 2x - 12 = 0 \]
Simplify:
\[ 2x^2 - 2x - 4 = 0 \]
Divide throughout by 2 for simplicity:
\[ x^2 - x - 2 = 0 \]
Factorize the quadratic equation:
\[ (x - 2)(x + 1) = 0 \]
Solve for x:
\[ x = 2 \]
\[ x = -1 \]
Therefore, the points of intersection are \( x = 2 \) and \( x = -1 \).
These points will be crucial in determining the limits of integration for setting up the volume integral in the subsequent steps.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad6d3305-3f03-4e1d-b5fa-fcc90cc2dc7c%2Fb3f74ef6-d866-4af6-9ef0-f006662080a1%2F5canycm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Volume of a Solid Generated by Revolving a Region Around the x-axis
To find the volume of the solid generated by revolving the region bounded by the graphs of the equations around the x-axis, follow these steps:
Given equations:
\[ y = x^2 + 8 \]
\[ y = -x^2 + 2x + 12 \]
\[ x = 0 \]
\[ x = 3 \]
#### Step 1:
**To find the point(s) of intersection of the curves \( y = x^2 + 8 \) and \( y = -x^2 + 2x + 12 \).**
We equate the two equations and solve for x.
\[ x^2 + 8 = -x^2 + 2x + 12 \]
Rearrange all terms to one side of the equation for standard quadratic form:
\[ x^2 + 8 + x^2 - 2x - 12 = 0 \]
Simplify:
\[ 2x^2 - 2x - 4 = 0 \]
Divide throughout by 2 for simplicity:
\[ x^2 - x - 2 = 0 \]
Factorize the quadratic equation:
\[ (x - 2)(x + 1) = 0 \]
Solve for x:
\[ x = 2 \]
\[ x = -1 \]
Therefore, the points of intersection are \( x = 2 \) and \( x = -1 \).
These points will be crucial in determining the limits of integration for setting up the volume integral in the subsequent steps.
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