   Chapter 5.5, Problem 23E ### 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 ) = x 3 + 4 x 2 , g ( x ) = x + 4

To determine

To graph: The region bounded by the graphs of f(x)=x3+4x2 and g(x)=x+4, and also compute the area of the region.

Explanation

Given Information:

The region bounded by the graphs of f(x)=x3+4x2 and g(x)=x+4.

Graph:

Consider the provided functions,

f(x)=x3+4x2 and g(x)=x+4

Consider the first function f(x)=x3+4x2

Compute the x-intercepts of the function,

x3+4x2=0x2(x+4)=0x=0 and 4

The x-intercepts are x=0, 4.

Compute the y-intercepts of the function,

y=03+4×02=0

The y-intercept is y=0.

Consider the second function g(x)=x+4

It is a linear function. So, its graph will be a straight line with slope m=1 and y-intercept b=4

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

f(x)=g(x)x3+4x2=x+4x3+4x2x4=0x2(x+4)1(x+4)=0

Simplify further,

(x+4)(x21)=0(x+4)(x1)(x+1)=0x=4,1 and 1

Substitute x=1 in the function g(x)=x+4 and compute the first intersection point,

g(1)=1+4=5

Substitute x=1 in the function g(x)=x+4 and compute the second intersection point,

g(1)=1+4=3

Substitute x=4 in the function g(x)=x+4 and compute the third intersection point,

g(4)=4+4=0

So, the graphs of f and g intersect at the points (4,0), (1,3) and (1,5). These three points give two intervals of integration

[4,1] and [1,1].

Sketch the graph of two functions as follows:

Formula used:

Area of a region bounded by two graphs is calculated using the following formula,

If f and g are continuous on [a,b] and g(x)f(x) for all x in [a,b], then the area of the region bounded by the graphs of f, g, x=a and x=b is given by

A=ab[f(x)g(x)]dx

The integral of xndx=xn+1n+1+C.

Calculation:

From the graph, g(x)f(x) for all x in the interval [4,1] and f(x)g(x) for all x in the interval [1,1]. Thus, use two integrals to determine the area of the region bounded by the graphs of f and g, one for interval [4,1] and one for the interval [1,1]

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