Problem 25 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

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 1: 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 \(e^{0.5x} = -\frac{1}{x}\).

Step 2: 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.

Step 3: 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.

Step 4: 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 \(x = 2\).

Step 5: 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.