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Use the shell method to find the volume of the solid generated following lines. a. The line x=5 c. The x-axis revolving the region bounded by the line y = 3x + 10 and the parabola y=x2 about the b. The line x = -2 d. The line y = 25

Use the shell method to find the volume of the solid generated following lines. a. The line x=5 c. The x-axis revolving the region bounded by the line y = 3x + 10 and the parabola y=x2 about the b. The line x = -2 d. The line y = 25

Advanced Engineering Mathematics
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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### Volume of Solids Using the Shell Method In this example, we will use the shell method to find the volume of the solid generated by revolving the region bounded by the line \( y = 3x + 10 \) and the parabola \( y = x^2 \) about various lines. #### Problem Statement Use the shell method to find the volume of the solid generated by revolving the region bounded by the line \( y = 3x + 10 \) and the parabola \( y = x^2 \) about the following lines: a. The line \( x = 5 \) b. The line \( x = -2 \) c. The x-axis d. The line \( y = 25 \) ### Step-by-Step Solution **Note:** Assume the graphs of the functions \( y = 3x + 10 \) and \( y = x^2 \) intersect to form a bounded region. 1. **Identify the region:** - Plot the line \( y = 3x + 10 \) and the parabola \( y = x^2 \) on the Cartesian plane. - Find the points of intersection, which are the solutions to \( 3x + 10 = x^2 \). 2. **Revolve the region:** - For each specified line, set up the integral using the shell method: - Determine the radius of the shell (distance to the axis of rotation). - Determine the height of the shell (difference between the two functions). 3. **Integrate:** - Integrate the expression over the interval defined by the points of intersection to find the volume. Each case \( (a, b, c, \text{and} d) \) will have a distinct axis of revolution, changing the radius and height of the shells accordingly. ### Case Analyses #### (a) Revolve about the line \( x = 5 \) - Radius: \( |5 - x| \) - Height: \( |3x + 10 - x^2| \) #### (b) Revolve about the line \( x = -2 \) - Radius: \( |-2 - x| \) - Height: \( |3x + 10 - x^2| \) #### (c) Revolve about the x-axis - Radius for shell method (using \( dy \) integration):
Transcribed Image Text:### Volume of Solids Using the Shell Method In this example, we will use the shell method to find the volume of the solid generated by revolving the region bounded by the line \( y = 3x + 10 \) and the parabola \( y = x^2 \) about various lines. #### Problem Statement Use the shell method to find the volume of the solid generated by revolving the region bounded by the line \( y = 3x + 10 \) and the parabola \( y = x^2 \) about the following lines: a. The line \( x = 5 \) b. The line \( x = -2 \) c. The x-axis d. The line \( y = 25 \) ### Step-by-Step Solution **Note:** Assume the graphs of the functions \( y = 3x + 10 \) and \( y = x^2 \) intersect to form a bounded region. 1. **Identify the region:** - Plot the line \( y = 3x + 10 \) and the parabola \( y = x^2 \) on the Cartesian plane. - Find the points of intersection, which are the solutions to \( 3x + 10 = x^2 \). 2. **Revolve the region:** - For each specified line, set up the integral using the shell method: - Determine the radius of the shell (distance to the axis of rotation). - Determine the height of the shell (difference between the two functions). 3. **Integrate:** - Integrate the expression over the interval defined by the points of intersection to find the volume. Each case \( (a, b, c, \text{and} d) \) will have a distinct axis of revolution, changing the radius and height of the shells accordingly. ### Case Analyses #### (a) Revolve about the line \( x = 5 \) - Radius: \( |5 - x| \) - Height: \( |3x + 10 - x^2| \) #### (b) Revolve about the line \( x = -2 \) - Radius: \( |-2 - x| \) - Height: \( |3x + 10 - x^2| \) #### (c) Revolve about the x-axis - Radius for shell method (using \( dy \) integration):
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