   Chapter 5.5, Problem 21E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Finding the Area Bounded by Two Graphs In Exercises 15-30, sketch the region bounded by the graphs of the functions and find the area of the region. See Examples 1, 2, 3, and 4. f ( x ) = 1 − x , g ( x ) = x 4 − x

To determine

To graph: The region bounded by the graphs of f(x)=1x and g(x)=x4x, and to compute the area of the region.

Explanation

Given Information:

The region bounded by the graphs of f(x)=1x and g(x)=x4x.

Graph:

Consider the provided functions,

f(x)=1x and g(x)=x4x

Consider the first function f(x)=1x

It is a linear function. So, its graph will be a straight line with slope m=1 and y-intercept b=1

Consider the second function g(x)=x4x

Compute the x-intercepts of the function,

x4x=0x(x31)=0x(x1)(x2+x+1)=0

The x-intercepts are x=0 and x=1.

Compute the points of intersection of two graphs by setting the functions equal to each other and solving for x,

f(x)=g(x)1x=x4xx4=1x=1, 1

Substitute x=1 in the function f(x)=1x and compute the first intersection point,

f(1)=11=0

Substitute x=1 in the function f(x)=1x and compute the second intersection point,

f(1)=1(1)=2

So, the graphs of given functions intersect at the points (1,0) and (1,2). These points give interval of integration [1,1].

Sketch the graph of two functions as follows:

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