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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 ) = x 3 , g ( x ) = x

To determine

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

Explanation

Given Information:

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

Graph:

Consider the provided functions,

f(x)=x3 and g(x)=x

Consider the first function f(x)=x3

Compute the x-intercepts of the function,

=x3=0x=03=0

The x-intercept is x=0.

As x increases, y increases and vice-versa. The graph will lie in the first and third quadrants.

Consider the second function g(x)=x

It is a linear function. So, its graph will be a straight line with slope m=1 and y-intercept b=0

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=xx=x3x3x=0

Simplify further,

x(x21)=0x(x1)(x+1)=0x=0,1 and 1

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

g(1)=1

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

g(1)=1

Substitute x=0 in the function g(x)=x and compute the third intersection point,

g(0)=0

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

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

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