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Find the volume of the solid obtained by rotating the region bounded by y = x, the line x = 4 and the x-axis about the line 2. Round your answer to two decimal places. Y =

Find the volume of the solid obtained by rotating the region bounded by y = x, the line x = 4 and the x-axis about the line 2. Round your answer to two decimal places. Y =

Calculus: Early Transcendentals
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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**Problem Statement:** Find the volume of the solid obtained by rotating the region bounded by \( y = \frac{1}{2}x \), the line \( x = 4 \), and the x-axis about the line \( y = 2 \). Round your answer to two decimal places. **Diagram Explanation:** This is a typical problem involving the application of the disk or washer method to find volumes of solids created by revolving regions around specific lines. 1. **Boundaries:** - The line \( y = \frac{1}{2}x \) is a linear function, representing a straight line through the origin with a slope of \( \frac{1}{2} \). - The vertical line \( x = 4 \) forms the right boundary of the region. - The x-axis (where \( y = 0 \)) serves as the lower boundary. 2. **Rotation Axis:** - The line \( y = 2 \) is a horizontal line parallel to the x-axis, and it serves as the axis of rotation. Using these boundaries and the axis of rotation, you can apply the appropriate methods to calculate the volume of the solid by considering the horizontal distance between the curve and the axis of rotation.
Transcribed Image Text:**Problem Statement:** Find the volume of the solid obtained by rotating the region bounded by \( y = \frac{1}{2}x \), the line \( x = 4 \), and the x-axis about the line \( y = 2 \). Round your answer to two decimal places. **Diagram Explanation:** This is a typical problem involving the application of the disk or washer method to find volumes of solids created by revolving regions around specific lines. 1. **Boundaries:** - The line \( y = \frac{1}{2}x \) is a linear function, representing a straight line through the origin with a slope of \( \frac{1}{2} \). - The vertical line \( x = 4 \) forms the right boundary of the region. - The x-axis (where \( y = 0 \)) serves as the lower boundary. 2. **Rotation Axis:** - The line \( y = 2 \) is a horizontal line parallel to the x-axis, and it serves as the axis of rotation. Using these boundaries and the axis of rotation, you can apply the appropriate methods to calculate the volume of the solid by considering the horizontal distance between the curve and the axis of rotation.
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