Problem 22 - 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. $$ y=\frac{e^{1 / x}}{x^{2}}, y=0, x=1, x=3 $$

Step-by-Step Solution

Answer

The area of the region bounded by the given curves is \(e^{1/3} - e\).

Step 1: Plot the Functions

It's important to get a visual understanding of the problem. By plotting the given function \(y=\frac{e^{1 / x}}{x^{2}}\) and the lines \(y=0\), \(x=1\), and \(x=3\), we can observe the region whose area we seek. This region is enclosed by the four bounded entities.

Step 2: Setup the Integral

Since we are looking for the area bounded by the curves, we set up an integral from 1 to 3 of the function \(\frac{e^{1 / x}}{x^{2}}\) with respect to \(x\). The integral will be \(\int_1^3 \frac{e^{1 / x}}{x^{2}} dx\).

Step 3: Solve the Integral

We make use of integral calculus to solve the definite integral. Substitution is our best bet in solving this integral. Let's set \(u = 1/x\), then \(du = -1/x^2 dx\) or \( - du = 1/x^2 dx\). Our integral will be \(- \int_{1/3}^{1} e^{u} du\). Now we perform the integration and we get \(-[e^u]_{1/3}^{1}\).

Step 4: Evaluate the Definite Integral

Evaluating the integral we get \(-[e - e^{1/3}]\).

Step 5: Simplify the Result

After simplifying, we get the area between the curves as \(e^{1/3} - e\).