   Chapter 5.5, Problem 20E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### 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 2 − x − 6 , y = 0 , x = − 2 , x = 1

To determine

To graph: The region bounded by the graphs of y=x2x6, y=0, x=2 and x=1, and to compute the area of the region.

Explanation

Given Information:

The region bounded by the graphs of y=x2x6, y=0, x=2 and x=1.

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

The integral of xndx=xn+1n+1+C.

Graph:

Consider the provided functions,

y=x2x6

And, y=0

Consider the first function y=x2x6.

It is a quadratic function. So, its graph will be a parabola with opening upwards. Factor the function,

g(x)=x2x6=x2+2x3x6=x(x+2)3(x+2)=(x+2)(x3)

The x-intercepts are x=2,3.

Consider the second function y=0

It is the equation of x-axis.

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

The integral of xndx=xn+1n+1+C.

Calculation:

From the graph, x2x60 for all x in the interval [2,1]

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