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3. Suppose R is the region in the plane bounded below by the curve y = x² and above by the line y = 1. a) Sketch R. Set up and evaluate an integral that gives the area of R. b) Suppose a solid has base R and the cross-sections of the solid perpendicular to the y-axis are squares. Sketch the solid and find its volume. c) Suppose a solid has base R and the cross-sections of the solid perpendicular to the y-axis are equilateral triangles. Sketch the solid and find its volume.

3. Suppose R is the region in the plane bounded below by the curve y = x² and above by the line y = 1. a) Sketch R. Set up and evaluate an integral that gives the area of R. b) Suppose a solid has base R and the cross-sections of the solid perpendicular to the y-axis are squares. Sketch the solid and find its volume. c) Suppose a solid has base R and the cross-sections of the solid perpendicular to the y-axis are equilateral triangles. Sketch the solid and find its volume.

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter10: Analytic Geometry
Section10.1: The Rectangular Coordinate System
Problem 41E: Find the exact lateral area of each solid in Exercise 40. Find the exact volume of the solid formed...
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3. Suppose R is the region in the plane bounded below by the curve y = x² and above by the line y = 1. a) Sketch R. Set up and evaluate an integral that gives the area of R. b) Suppose a solid has base R and the cross-sections of the solid perpendicular to the y-axis are squares. Sketch the solid and find its volume. c) Suppose a solid has base R and the cross-sections of the solid perpendicular to the y-axis are equilateral triangles. Sketch the solid and find its volume. 4. Start with the region A in the first quadrant enclosed by the x-axis and the parabola y = 2x(2-x). Then obtain solids of revolution S₁, S2, and S3 by revolving A about the lines y = 4, y = -2, and x = 4 respectively. All three solids are (unusual) "doughnuts" which are 8 units across, whose hole is 4 units across, and whose height is 2 units. Sketch them. a) Which do you expect to have larger volume, S₁ or S₂? Compute their volumes exactly and check your guess. b) Compute the volume of S3. (It may be harder to guess in advance how S3 compares in volume to S2 and S₁.) 1
Transcribed Image Text:3. Suppose R is the region in the plane bounded below by the curve y = x² and above by the line y = 1. a) Sketch R. Set up and evaluate an integral that gives the area of R. b) Suppose a solid has base R and the cross-sections of the solid perpendicular to the y-axis are squares. Sketch the solid and find its volume. c) Suppose a solid has base R and the cross-sections of the solid perpendicular to the y-axis are equilateral triangles. Sketch the solid and find its volume. 4. Start with the region A in the first quadrant enclosed by the x-axis and the parabola y = 2x(2-x). Then obtain solids of revolution S₁, S2, and S3 by revolving A about the lines y = 4, y = -2, and x = 4 respectively. All three solids are (unusual) "doughnuts" which are 8 units across, whose hole is 4 units across, and whose height is 2 units. Sketch them. a) Which do you expect to have larger volume, S₁ or S₂? Compute their volumes exactly and check your guess. b) Compute the volume of S3. (It may be harder to guess in advance how S3 compares in volume to S2 and S₁.) 1
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