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Chapter 11: Problem 25

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(x)=e^{0.5 x}, g(x)=-\frac{1}{x}, x=1, x=2 $$

Short Answer

Expert verified
The area of the region bounded by the functions \(e^{0.5 x}\), \(-\frac{1}{x}\) between \(x=1\) and \(x=2\) is found by setting up integrations using these functions, calculating them and adding the results. The specific value of the area depends on the point of intersection \(a\).

Step by step solution

01

Identify the Intersecting Point

First, it is important to identify the point where the functions intersect in the interval [1, 2]. This can be done by setting \(f(x) = g(x)\) and solving for \(x\). Find the common point \(a\) such that \(e^{0.5x} = -\frac{1}{x}\).
02

Graph the Functions

Once the intersecting point is identified, the next step is to draw the curves for the functions in the defined range ([1, 2]) to visually understand the problem. Try to graph \(f(x)\) and \(g(x)\) and shade the area to be calculated.
03

Set Up the Integration

To calculate the area, formulate an integration from \(x=1\) to \(x=a\) for \(f(x) - g(x)\), and from \(x=a\) to \(x=2\) for \(g(x) - f(x)\). Since the some part of \(g(x)\) is less than \(f(x)\) and some part of it is greater than \(f(x)\), we calculate the area in two parts.
04

Evaluating the Integration

Perform the integration and add the results of the two integrals. That sum will represent the total area bounded by the two functions between \(x = 1\) and \(x = 2\).
05

Interpret the Calculated Area

After completing the computation, it is important to understand its implications. Interpret the final result and review any assumptions or caveats about the found area.

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Most popular questions from this chapter

The revenue from a manufacturing process (in millions of dollars per year) is projected to follow the model \(R=100\) for 10 years. Over the same period of time, the cost (in millions of dollars per year) is projected to follow the model \(C=60+0.2 t^{2}\), where \(t\) is the time (in years). Approximate the profit over the 10 -year period.

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