   Chapter 13, Problem 22RE ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Find the area between the curves in Problems 19-22. y = x 3 − 1  and  y = x − 1

To determine

To calculate: The area of the region between the curve y=x31 and y=x1.

Explanation

Given information:

The provided curve is y=x31 and y=x1.

Formula used:

Area between two curves,

For the continuous function f and g on [a,b] and if f(x)g(x) on [a,b], then the area of the region bounded by y=f(x), y=g(x), x=a, and x=b is:

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

Or

A=Integral of (Top Bottom)

Calculation:

Consider the provided curves,

y=x31

And,

y=x1

Draw the graph of the above function to check whether f(x)g(x):

Calculate the values at different values of x:

Substitute x=0 in the function y=x31,

y=(0)31=1

Substitute x=1 in the function y=x31,

y=(1)31=11=0

Substitute x=2 in the function y=x31,

y=(2)31=7

Substitute x=1 in the function y=x31,

y=(1)31=11=2

The table provided below shows the values of the function at different values of x,

 x y=x3−1 Coordinates (x,y) 0 −1 (0,−1) 1 0 (1,0) 2 7 (2,7) −1 −2 (−1,−2)

Now, consider y=x1,

Substitute x=0 in the function y=x1.

y=01=1

Substitute x=1 in the function y=x1.

y=11=0

Substitute x=2 in the function y=x1.

y=21=1

Substitute x=1 in the function y=x1.

y=11=2

The table provided below shows the values of the function at different values of x,

 x y=x−1 Coordinates (x,y) 0 −1 (0,−1) 1 0 (1,0) 2 1 (2,1) −1 −2 (−1,−2)

Use the above tables and information to draw the graph of y=x31 and y=x1 in the same graph to check whether f(x)g(x):

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