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The definite integral is a fundamental concept in calculus that is used to calculate the net area under a curve between two points on the x-axis. In our exercise, we utilize the definite integral to find the area of a region bounded by a function and the x-axis.

To set up a definite integral for this problem, we determine the limits of integration, which are the x-values that form the boundaries of the region. Here, the limits are from 1 to 3, as specified in the problem. By performing the integral, we accumulate the infinitesimal quantities from the function over this interval and effectively figure out the area under the curve.

Thus, the calculation involves integrating the function from 1 to 3 to derive the total area.

In calculus, the area under a curve represents the region between the function's graph and the x-axis, within your specified limits. In the exercise, this means calculating the space between the curve defined by the function \(f(x) = \frac{1}{x^2} e^{1/x}\) and the x-axis from x = 1 to x = 3.

The process to find this area involves setting up and solving a definite integral, as described in the previous section. The result provides a numerical value that represents the total area enclosed by these boundaries.

This concept is crucial because it allows us to apply calculus practically, turning complicated wave-like forms into manageable calculations for areas and applications such as physics or engineering.

U-substitution is a vital technique in calculus used to simplify the integration process. It's particularly useful when dealing with composite functions, allowing us to convert a complex integral into a more straightforward form.

In this exercise, we apply u-substitution to tackle the integral \(\int_{1}^{3} \frac{1}{x^2} e^{1/x} \, dx\). Here, let \(u = \frac{1}{x}\), which leads to \(du = -\frac{1}{x^2} \, dx\). This substitution changes the problem into a simpler integral, \(-\int e^u \, du\), with new limits of 1 to 1/3.

Once simplified and evaluated, the integration becomes much easier to resolve, allowing us to effectively find the area under the curve.

A bounded region in this context refers to the area enclosed by the curve of the function, the x-axis, and the vertical lines at specific x-values. In our exercise, the bounded region lies between the curve \(f(x) = \frac{1}{x^2} e^{1/x}\), the x-axis (y = 0), and from x = 1 to x = 3.

This gives us a finite area to calculate, as the curve and these lines form a closed shape on the graph.

• Upper Boundary: The curve given by the function \(f(x)\).
• Lower Boundary: The x-axis (y = 0).
• Vertical Boundaries: Lines x = 1 and x = 3.

Understanding these boundaries is crucial because they define the limits of your integration and specify exactly what "area under the curve" you're considering.