   Chapter 5.5, Problem 26E ### 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 = e 1 / x x 2 , y = 3 , x = 1 , x = 3

To determine

To graph: The region bounded by the graphs of y=e1xx2, y=3, x=1 and x=3, and also find the area of the region.

Explanation

Given Information:

The region bounded by the graphs of y=e1xx2, y=3, x=1 and x=3.

Graph:

Consider the following functions,

y=e1xx2, y=3, x=1 and x=3

Consider the first function y=e1xx2

It is an exponential function with positive exponent. So, its graph will be similar to the graph f exponential function for positive x.

Compute the x-intercepts of the function,

e1xx2=0e1x=0   (not possible)

So, there is no x-intercept.

Compute the y-intercepts of the function,

y=e1002   (not possible)

So, there is no y-intercept.

Consider the second function y=3

It is the equation of a horizontal line with y-intercept y=3

Sketch the graph of two functions as follows:

Formula used:

The derivative of d(xn)dx=nxn1.

The integration of xndx=xn+1n+1+c and exdx=ex+c.

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

Calculation:

From the graph, e1xx23 for all x in the interval [1,3]. Assume f(x)=3 and g(x)=e1xx2, use the formula for the area to calculate the area of the provided region,

Area=13[f(x)g(x)]dx=13[3e1xx2]dx=133dx13e1xx2dx

Assume that u=1x, then dudx=1x2

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