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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 5.5, Problem 4CP
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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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Checkpoint 4 Worked-out solution available at LarsonAppliedCalculus.com

Find the area of the region bounded by the graphs of

f ( x ) = x 3 + 2 x 2 − 3 x  and g ( x ) = x 2 + 3 x

Sketch a graph of the region.

To determine

To calculate: The area of the region bounded by the graphs of f(x)=x3+2x2−3x and g(x)=x2+3x and then sketch the graph.

Explanation

Given Information:

The two functions f(x)=x3+2x2−3x and g(x)=x2+3x.

Formula used:

The 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 power rule is ∫xn⋅dx=xn+1n+1+C,n≠−1.

Calculation:

Consider the provided functions,

f(x)=x3+2x2−3x

And, g(x)=x2+3x

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+2x2−3x=x2+3xx3+2x2−x2−3x−3x=0x3+x2−6x=0

Factor the above polynomial,

x(x2+x−6)=0x(x2+3x−2x−6)=0x[x(x+3)−2(x+3)]=0x(x+3)(x−2)=0

That is,

x=0x+3=0x=−3

And,

x−2=0x=2

Substitute x=−3 in the function f(x)=x3+2x2−3x and compute the first intersection point,

f(−3)=(−3)3+2×(−3)2−3×(−3)=−27+18+9=0

Substitute x=0 in the function f(x)=x3+2x2−3x and compute the second intersection point,

f(0)=03+2×02−3×0=0+0−0=0

Substitute x=2 in the function f(x)=x3+2x2−3x and compute the third intersection point,

f(2)=23+2×22−3×2=8+8−6=10

So, the graphs of f and g intersect at the points (−3,0), (0,0) and (2,10). These three points give two intervals of integration.

Sketch the graph of two functions as follows:

From the graph, g(x)≤f(x) for all x in the interval [−3,0] and f(x)≤g(x) for all x in the interval [0,2]

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Mathematics For Machine Technology

Exercises 4652 refer to the following: A nonprofit organizations board of directors, composed of four women Ang...

Mathematics: A Practical Odyssey

Solve each system by any method. 3x+y+z=02x-y+z=02x+y+z=0

College Algebra (MindTap Course List)

How large a sample should be selected to provide a 95% confidence interval with a margin of error of 10? Assume...

Statistics for Business & Economics, Revised (MindTap Course List)

An automobile dealer conducted a test to determine if the time in minutes needed to complete a minor engine tun...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

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