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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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In Exercises 7–10, find the area of the region bounded by the graphs of f and g.

f ( x ) = x 3 3 x 2 + 2 , g ( x ) = x 1

To determine

To calculate: The area of the region bounded by the graphs of f(x)=x33x2+2 and g(x)=x1.

Explanation

Given Information:

The functions are f(x)=x33x2+2 and g(x)=x1.

Formula used:

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:

Consider the function, f(x)=x33x2+2

Let y=f(x)

y=x33x2+2

Now, in order to plot the graph choose some values for x and calculate the value for y to form the ordered pairs that will plot on the graph.

So, put x=0 in the equation y=x33x2+2, then, y=2

Now, put x=1,

y=133×12+2=33=0

Now, put x=2

y=233×22+2=812+2=2

The required table is shown below,

x y (x,y)
0 2 (0,2)
1 0 (1,0)
2 2 (2,2)

So, its graph appears as follows:

Consider the second function, g(x)=x1

Let y=f(x), then

y=x1

For x=0, y=4×0=0 and

For x=5, y=4×(5)=20

It represents a straight line passing through points.

x 0 1
y -1 0

To plot them together, calculate the point of intersection of y=x33x2+2 and y=x1,

Equating the y of both the equation and solve for x,

x33x2+2=x1x33x2x+3=0x2(x1)2x(x1)3(x1)=0(x1)(x3)(x+1)=0

Equating each of the factor to zero, the values x are x=1, x=1and x=3,

For x=1, y=0 and

For x=1, y=2

For x=3, y=2

Thus, the point of the intersection are (1,2),(1,0)and (3,2)

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