   Chapter 5.R, Problem 16E

Chapter
Section
Textbook Problem

# Let ℜ be the region in the first quadrant bounded by the curves y = x 3 and y = 2 x − x 2 . Calculate the following quantities.(a) The area of ℜ (b) The volume obtained by rotating ℜ about the x -axis(c) The volume obtained by rotating ℜ about the y -axis

To determine

a)

To calculate:

The area of R

Explanation

1) Concept:

The area A of the region bounded by the curves y=f(x), y=g(x) and the lines x=a and x=b, where f and g are continuous and fxgx for all x in a, b, is

A= abfx-gxdx

2) Given:

The equations of the curvesare y=x3,y=2x-x2.

3) Calculation:

First find the intersection points of the curves by solving their equations simultaneously.

x3=2x-x2

x3+x2-2x=0

xx2+x-2=0

x(x+2)(x-1)=0

x=0, x=-2 and x=1

Since the region lies in the first quadrant, only x=0  and  x=1 are the required values

To determine

b)

To calculate:

The volume obtained by rotating R  about the x  axis.

To determine

c)

To calculate:

The volume obtained by rotating R  about y  axis.

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