Question 2 (40 points): Consider the region R bounded by y = x² and y=x. (a) (10 points) Sketch the region R and find the area of the region R. (b) (15 points) Find the volume of the solid obtained by revolving the region R about the line y = 1 using disk-washer method. (c) (15 points) Find the volume obtained by revolving the region R about the x = 1 using cylindrical-shell method. Solution: (a) (0.0) (1.1) Find the intersection points: (0,0) and (1,1). Area of R 04 " 1 LE 5 points = √ ²( x − x² ) d x = ( ² ² = ²) | = ¦¦- (b) 2 3 16 Take a vertical slice at x with thickness dx and rotate it about the y = 1. Then volume of the washer is dV = [(1 − x²)² - π(1-x)²]dx = n(x² - 3x² + 2x)dx V = √ dV = √(x² -3x² + 2x)dx 5 points ------- 10 points - 4 points =(-1+1)= (c) Take a vertical slice at x with thickness dx, rotate it about the line x = 1 point x= 1. It forms a cylindrical shell. Its volume is given by dV = 2(1-x)(x − x²)dx v=fdv=2(1-x)(x - x²)dx 10 points

Question 2 (40 points): Consider the region R bounded by y = x² and y=x. (a) (10 points) Sketch the region R and find the area of the region R. (b) (15 points) Find the volume of the solid obtained by revolving the region R about the line y = 1 using disk-washer method. (c) (15 points) Find the volume obtained by revolving the region R about the x = 1 using cylindrical-shell method. Solution: (a) (0.0) (1.1) Find the intersection points: (0,0) and (1,1). Area of R 04 " 1 LE 5 points = √ ²( x − x² ) d x = ( ² ² = ²) | = ¦¦- (b) 2 3 16 Take a vertical slice at x with thickness dx and rotate it about the y = 1. Then volume of the washer is dV = [(1 − x²)² - π(1-x)²]dx = n(x² - 3x² + 2x)dx V = √ dV = √(x² -3x² + 2x)dx 5 points ------- 10 points - 4 points =(-1+1)= (c) Take a vertical slice at x with thickness dx, rotate it about the line x = 1 point x= 1. It forms a cylindrical shell. Its volume is given by dV = 2(1-x)(x − x²)dx v=fdv=2(1-x)(x - x²)dx 10 points

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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