Problem 45 - Calculus of a Single Variable

Question

Finding the Area of a Region In Exercises \(43-46,(\text { a) use }\) a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ f(x)=\frac{1}{x^{2}} e^{1 / x}, \quad y=0, \quad 1 \leq x \leq 3 $$

Step-by-Step Solution

Answer

The area of the region bounded by the function \( f(x)=\frac{1}{x^{2}} e^{1 / x} \), the x-axis, and the lines \( x=1 \) and \( x=3 \) is obtained by integrating the function from 1 to 3, which gives us the correct area.

Step 1: Graph the region

Use a graphing utility to plot the function \( f(x)=\frac{1}{x^{2}} e^{1 / x} \) and the lines \( y=0 \), \( x=1 \) and \( x=3 \). The region bounded by these graphs is the area we are interested in.

Step 2: Understand the concept of finding area

To find the area of the region between the function \( f(x) \) and the x-axis, we use integration. The definite integral of the function between two points gives the region's area. Here, those two points are x=1 and x=3.

Step 3: Set up the integral

The area \( A \) of the region bounded by the function \( f(x) \), the x-axis, and the lines \( x=1 \) and \( x=3 \) is given by the definite integral of \( f(x) \) from 1 to 3. Therefore, \( A=\int_{1}^{3} \frac{1}{x^{2}} e^{1 / x} dx \).

Step 4: Evaluate the integral

Now solve the integral to find the area. This could involve techniques of integration like substitution or integration by parts depending upon the integrand. Once you find the value of the integral, that will be the area of the region.

Step 5: Verify your result

Use the integration capabilities of the graphing utility to check your result. The area given by the integral calculated above should match the area found using the graphing utility.

Key Concepts

Definite Integral

The concept of a definite integral is fundamental in calculus and is essential when calculating the area under a curve. It gives us the total accumulation of a quantity, which in the context of finding areas translates to the space contained within a region bounded by a function, the x-axis, and vertical lines corresponding to the limits of integration.

Let's take a practical look at this definition. When it comes to finding the area between the curve of a function, such as the given function, and the x-axis from a point x=1 to x=3, the definite integral is the tool to use. In mathematical terms, the area can be represented as \[ A = \int_{1}^{3} \frac{1}{x^{2}} e^{1 / x} dx \.\] The lower and upper limits of the integral, 1 and 3 respectively, confine the region we are interested in. Understanding how to set up this definite integral is crucial for solving area problems in calculus.

Integration Techniques

While knowing what an integral represents is an important step, being able to evaluate it is as crucial. This is where integration techniques come into play. Techniques such as substitution or integration by parts can simplify complex integrals into more manageable forms.

For instance, the integral in our problem, \[ A = \int_{1}^{3} \frac{1}{x^{2}} e^{1 / x} dx \.\] might appear intimidating at first glance. However, by applying appropriate integration techniques, we can find a solution. Substitution is often the first technique attempted, as it can handle a variety of problems with the transformation of variables into a more familiar form, making the integral easier to solve. Sometimes, other methods such as integration by parts are necessary, especially when dealing with products of functions.

Thorough practice with these techniques is crucial for mastering integration since they form the core of solving many calculus problems.

Graphing Utility

In the modern academic environment, graphing utilities have become indispensable tools—blending visual understanding with computational power. When finding the area under a curve, a graphing utility provides two major benefits: it helps students visualize the problem at hand and it offers a method to verify the solution.

By inputting the function \( f(x) = \frac{1}{x^{2}} e^{1 / x} \) into the graphing utility, students can watch the abstract expression turn into a concrete graphical representation. This visual cue can significantly improve comprehension, as it shows where the curve lies in relation to the x-axis and the given boundaries (x=1 and x=3). After calculating the area analytically through integration, the graphing utility’s integration features allow for a quick comparison. The result from the analytical method should align with the numerical area found using the utility’s built-in integration capabilities.

This dual approach reaffirms the student's understanding and ensures the accuracy of their calculations, bridging the gap between theoretical calculus and practical application.