Problem 43 - Essential Calculus

Question

(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

First, we plot the function \(f(x) = \frac{1}{x^{2}} e^{1/x}\) using a graphing utility. Then we compute the integral of this function over the interval [1,3]. Finally, we verify these manual findings using the integration function of the graphing utility. Note: The values from Steps 2 and 3 are not included here as they require a calculator or software for precise calculation.

Step 1: Graph the Function

Plot the function \(f(x) = \frac{1}{x^{2}} e^{1/x}\) using a graphing utility within the interval 1 ≤ x ≤ 3.

Step 2: Finding Area under the curve

To find the area under the curve, compute the definite integral of \(f(x)\) over the interval from 1 to 3. The integral is \(\int_{1}^{3}\frac{1}{x^{2}} e^{1/x} dx\).

Step 3: Verify the Results

Use the graphing utility's integration feature to compute the definite integral of the function from 1 to 3. Verify that the computational result matches the hand-calculated result.

Key Concepts

Definite Integral

The concept of a definite integral in calculus is crucial for calculating the area under a curve between two specified points on the x-axis. Unlike indefinite integrals, definite integrals yield a number, representing this area. For the given problem, we compute the definite integral of the function \(f(x) = \frac{1}{x^{2}} e^{1/x}\) over the interval \([1, 3]\). This is expressed mathematically as \(\int_{1}^{3} \frac{1}{x^{2}} e^{1/x} dx\). A definite integral takes into account both the function and the bounds \([a, b]\), thereby allowing for an exact calculation of the area enclosed. Understanding this concept is key to solving problems related to areas under curves and verifying results obtained via computational tools.

Area Under a Curve

Finding the area under a curve involves integrating the function over the desired interval. This concept is pivotal in calculus as it applies to numerous fields such as physics, economics, and engineering. The curve's area between the limits 1 and 3 can be visualized as the space between \(f(x) = \frac{1}{x^{2}} e^{1/x}\) and the x-axis. The process involves:

• Setting up the integral \(\int_{1}^{3} \frac{1}{x^{2}} e^{1/x} dx\).
• Calculating this integral using analytical methods or numerical approximations.

Such calculations help visually and numerically estimate the space occupied by the curve within any given interval, enabling practical applications and validations through computational tools.

Graphing Utility

Graphing utilities are incredibly useful tools in modern mathematics and education. They bridge the gap between manual calculations and technological visualization. Graphing the function \(f(x) = \frac{1}{x^{2}} e^{1/x}\) over \([1, 3]\) allows one to visually assess the region of interest. Tools such as graphing calculators or software like Desmos can assist in:

• Sketching accurate graphs of complex functions.
• Automating the calculation of definite integrals.
• Providing a visual comparison of theoretical and practical results.

Using a graphing utility not only verifies manual integration results by calculating areas but also enhances comprehension by offering an intuitive understanding of calculus concepts through visualization.