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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.

y = x 3 + 3 , y = x , x = 1 , x = 1

To determine

To graph: The region bounded by the graphs of y=x3+3, y=x, x=1 and x=1, and also compute the area of the region.

Explanation

Given Information:

The region bounded by the graphs of y=x3+3, y=x, x=1 and x=1.

Graph:

Consider the provided functions,

y=x3+3

And, y=x

Consider the first function y=x3+3

It is a cubic function. Compute its x-intercepts as follows:,

x3+3=0x3=3x=33

The x-intercept is x=33.

Compute its y-intercepts as follows:,

y=03+3=3

The y-intercept is y=3.

Consider the second function y=x

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

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

Calculation:

From the graph, x(x3+3) for all x in the interval [1,1]

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