91Ó°ÊÓ

Definite integrals are a fundamental concept in calculus that help us find the area under a curve over an interval. Consider it like summing up an infinite number of infinitesimally small areas to get the total area. This process involves setting up and evaluating an integral with specific limits, which are usually provided by the problem.
In our exercise, the curve is defined by the function \( f(x) = 2 \sec \left(\frac{\pi x}{6}\right) \) with the limits of integration being \( x = 0 \) and \( x = 2 \). The definite integral is written as:

• \( \int_{0}^{2} 2 \sec \left(\frac{\pi x}{6}\right) \, dx \)

The result of evaluating this definite integral will give us the area of the region bounded by these limits and the x-axis. It's important to understand that definite integrals account for both the magnitude and direction of areas, where areas below the x-axis are considered negative. Here, since the function remains positive, our final area will also be positive.

The area under a curve is a concept that roots deep in integral calculus, where the focus is on finding the size of the region bounded by the graph of a function and the x-axis over a given interval. This area is precisely what a definite integral calculates.
For the given problem, the function \( y = 2 \sec \left(\frac{\pi x}{6}\right) \) determines the height of the curve above the x-axis between \( x = 0 \) and \( x = 2 \). By integrating this function over the specified limits, we can calculate the total area beneath the curve.
This is especially useful in practical situations where direct measurement of area is not possible, such as finding the total distance traveled given a velocity-time graph.

The secant function, often abbreviated as "sec", is one of the primary trigonometric functions. It represents the reciprocal of the cosine function, defined as \( \sec \theta = \frac{1}{\cos \theta} \).
In our integration problem, the function we're integrating is \( y = 2 \sec \left(\frac{\pi x}{6}\right) \). The secant function takes the role of determining how our curve stretches vertically.
A deep understanding of the secant function is crucial, especially since it appears less frequently than sine or cosine, but brings with it distinct computational properties. In terms of integration, recalling its properties can simplify complex trigonometric integrals.

Trigonometric integrals involve the integration of functions that include trigonometric expressions. They often require special techniques or formulas to solve them.
For the problem at hand, the trigonometric integral involves \( \sec \left(\frac{\pi x}{6}\right) \), and the identity we used is:

• \( \int \sec x \, dx = \ln |\sec x + \tan x| + C \)

This formula helps us compute the integral by transforming a trigonometric expression into a logarithmic form. Here, an adjustment is needed for the argument \( \frac{\pi x}{6} \), resulting in:

• \( \int \sec \left(\frac{\pi x}{6}\right) \, dx = \frac{6}{\pi} \ln |\sec (\frac{\pi x}{6}) + \tan (\frac{\pi x}{6})| + C \)

Mastering the integration of such trigonometric functions can be particularly rewarding, as it opens up solutions to many problems involving oscillations, waves, and even electrical circuits.