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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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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 ) = 3 x + 1 , g ( x ) = x + 1

To determine

To graph: The region bounded by the graphs of f(x)=3x+1 and g(x)=x+1, and also compute the area of the region.

Explanation

Given Information:

The region bounded by the graphs of f(x)=3x+1 and g(x)=x+1.

Graph:

Consider the provided functions,

f(x)=3x+1 and g(x)=x+1

Consider the first function f(x)=3x+1

Compute the x-intercepts of the function,

3x+1=03x=13x=1x=13

The x-intercept is x=13.

Compute the y-intercepts of the function,

y=3×0+1=0+1=1

The y-intercept is y=1.

Consider the second function g(x)=x+1

It is a linear function. So, its graph will be a straight line with slope m=1 and y-intercept b=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)3x+1=x+13x=x3x=x2

Factor the polynomial,

x23x=0x(x3)=0x=0, 3

Substitute x=0 in the function f(x)=3x+1 and compute the first intersection point,

f(0)=3×0+1=0+1=1

Substitute x=3 in the function f(x)=3x+1 and compute the second intersection point,

f(3)=3×3+1=3+1=4

So, the graphs of given functions intersect at the points (0,1) and (3,4). These points give interval of integration [0,3].

Sketch the graph of two functions as follows:

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