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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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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.

f ( x ) = e 0.5 x , g ( x ) = 1 x , x = 1 , x = 2

To determine

To graph: The region bounded by the graphs of f(x)=e0.5x, g(x)=1x, x=1 and x=2, and also compute the area of the region.

Explanation

Given Information:

The region bounded by the graphs of f(x)=e0.5x, g(x)=1x, x=1 and x=2.

Graph:

Consider the following equations that give the required region,

f(x)=e0.5x, g(x)=1x, x=1 and x=2

Then the first function f(x)=e0.5x is an exponential function with positive exponent. So, its graph will be similar to the graph of an exponential function.

There will be no x-intercept.

Consider the second function g(x)=1x

It is a reciprocal function. The graph will lie in second and fourth quadrant. As x increases in both directions, y will decrease and vice-versa.

Sketch the graph of the region 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

Exponential and log rule for integrals are,

exdx=ex+C1xdx=ln|x|+C

Calculation:

From the graph, 1xe0.5x for all x in the interval [1,2]. Use the formula for the area to calculate the area of the provided region,

Then,

Area=12[f(x)g(x)]dx=12[e0.5x(1x)]dx=12[e0

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