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1 2. Let R be the region bounded by the curves y = x and y = x³. Let S be the solid generated when R is revolved about the x-axis in the first quadrant. Find the volume of S by both the disc/washer and shell methods. Check that your results agree. -0:5 0 0,5

1 2. Let R be the region bounded by the curves y = x and y = x³. Let S be the solid generated when R is revolved about the x-axis in the first quadrant. Find the volume of S by both the disc/washer and shell methods. Check that your results agree. -0:5 0 0,5

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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### Problem Statement: Let \( R \) be the region bounded by the curves \( y = x \) and \( y = x^{\frac{1}{3}} \). Let \( S \) be the solid generated when \( R \) is revolved about the \( x \)-axis in the first quadrant. Find the volume of \( S \) by both the disc/washer and shell methods. Check that your results agree. #### Graph Explanation: - The graph provided shows the region \( R \) highlighted in green. - The x-axis runs horizontally, and the y-axis runs vertically. - The curve \( y = x \) is the diagonal line running from the origin (0,0) to (1,1). - The curve \( y = x^{\frac{1}{3}} \) starts at the origin (0,0), quickly rises but at a decreasing rate as \( x \) approaches 1. - The area of interest is bounded between these two curves and the \( x \)-axis in the first quadrant. #### Visualization: - x-axis and y-axis intersect at the origin (0, 0). - Both curves intersect at the point (1, 1). - This creates a region in the shape of a curvilinear triangle with one side along the line \( y = x \), another along \( y = x^{\frac{1}{3}} \), and the x-axis as the base. #### Objective: To find the volume of \( S \) which is the solid formed by revolving region \( R \) around the x-axis. #### Instructions for Finding Volume: 1. **Disc/Washer Method**: - Identify the radius of the discs formed by revolving around the x-axis. - Integrate the squared radius difference over the bounds of \( x \) from 0 to 1. 2. **Shell Method**: - Calculate the volume of cylindrical shells formed by rotating around the x-axis. - Integrate the surface area of each shell over the range of y-values. Always ensure that the volume calculated by both methods matches as a verification step.
Transcribed Image Text:### Problem Statement: Let \( R \) be the region bounded by the curves \( y = x \) and \( y = x^{\frac{1}{3}} \). Let \( S \) be the solid generated when \( R \) is revolved about the \( x \)-axis in the first quadrant. Find the volume of \( S \) by both the disc/washer and shell methods. Check that your results agree. #### Graph Explanation: - The graph provided shows the region \( R \) highlighted in green. - The x-axis runs horizontally, and the y-axis runs vertically. - The curve \( y = x \) is the diagonal line running from the origin (0,0) to (1,1). - The curve \( y = x^{\frac{1}{3}} \) starts at the origin (0,0), quickly rises but at a decreasing rate as \( x \) approaches 1. - The area of interest is bounded between these two curves and the \( x \)-axis in the first quadrant. #### Visualization: - x-axis and y-axis intersect at the origin (0, 0). - Both curves intersect at the point (1, 1). - This creates a region in the shape of a curvilinear triangle with one side along the line \( y = x \), another along \( y = x^{\frac{1}{3}} \), and the x-axis as the base. #### Objective: To find the volume of \( S \) which is the solid formed by revolving region \( R \) around the x-axis. #### Instructions for Finding Volume: 1. **Disc/Washer Method**: - Identify the radius of the discs formed by revolving around the x-axis. - Integrate the squared radius difference over the bounds of \( x \) from 0 to 1. 2. **Shell Method**: - Calculate the volume of cylindrical shells formed by rotating around the x-axis. - Integrate the surface area of each shell over the range of y-values. Always ensure that the volume calculated by both methods matches as a verification step.
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