   Chapter 13.3, Problem 21E ### 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

# In Problems 13-26, equations are given whose graphs enclose a region. In each problem, find the area of the region. f ( x ) = x 3 ;   g ( x ) = x 2 + 2 x

To determine

To calculate: The area of the region between f(x)=x3 and g(x)=x2+2x.

Explanation

Given information:

The provided curve is:

f(x)=x3 and g(x)=x2+2x.

Formula used:

Area between two curve:

If f and g are continuous functions 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 equation:

f(x)=x3 and g(x)=x2+2x

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 f(x)=x3.

y=(0)3=0

Substitute x=1 in the function f(x)=x3.

y=(1)3=1

Substitute x=2 in the function f(x)=x3.

y=(2)3=8

Substitute x=1 in the function f(x)=x3.

y=(1)3=1

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

 x f(x)=x3 Coordinates (x,y) 0 0 (0,0) 1 1 (1,1) 2 8 (2,8) −1 −1 (−1,−1)

Now, consider g(x)=x2+2x

Substitute x=0 in the function g(x)=x2+2x.

y=(0)2+2(0)=0

Substitute x=1 in the function g(x)=x2+2x.

y=(1)2+2(1)=3

Substitute x=2 in the function g(x)=x2+2x.

y=(2)2+2(2)=8

Substitute x=1 in the function g(x)=x2+2x.

y=(1)2+2(1)=1

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

 x g(x)=x2+2x Coordinates (x,y) 0 0 (0,0) 1 3 (1,3) 2 8 (2,8) −1 −1 (−1,−1)

Use the above tables and information to draw the graph of f(x)=x3 and g(x)=x2+2x in the same graph so as to check whether f(x)g(x):

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