91Ó°ÊÓ

When we talk about finding the area under a curve, we are looking at the region that lies between the curve and the x-axis, within certain boundary values of x.
This is a common requirement in calculus to understand the cumulative value that a function represents over an interval.
In this particular problem, our aim is to determine how much area is enclosed by the curve of the function \( f(x) = \frac{1}{x} \) from \( x = 1 \) to \( x = 5 \).

• Think of the area as a collection of infinitely thin rectangles spanning from \( x = 1 \) to \( x = 5 \).
• Each rectangle's height is defined by the function \( f(x) = \frac{1}{x} \), and its width is an incredibly tiny bit of the distance on the x-axis.

By summing up the areas of all these tiny rectangles, you get the total area under the curve. This summation is precisely what a definite integral calculates. This area gives us a tangible numerical value that we can use in practical applications.

An antiderivative, also known as an indefinite integral, is a function whose derivative matches the original function you started with.
In simple terms, it is the "reverse" of taking a derivative.
For our function, \( f(x) = \frac{1}{x} \), the antiderivative is \( \ln|x| \).

• The role of the antiderivative is crucial because it allows us to express the accumulation of area under the curve analytically.
• To find the area under \( f(x) = \frac{1}{x} \) from \( x = 1 \) to \( x = 5 \), we use this antiderivative.

Why logarithms? The natural logarithm (ln) is closely linked to the function \( \frac{1}{x} \) through differentiation rules. Therefore, \( \ln|x| \) correctly captures the accumulation function when dealing with \( \frac{1}{x} \).

Definite integration provides us with a structured approach to finding the area under a curve between two bounds on the x-axis. Let's break down the process into clear steps when evaluating a definite integral:

• **Setting Up the Integral**: Start by identifying the integral expression with correct limits of integration, like \( \int_{1}^{5} \frac{1}{x} \, dx \) for our exercise.
• **Determining the Antiderivative**: You've found the antiderivative as \( \ln|x| \). This function acts as a tool to compute accumulated change over the interval.
• **Apply the Limits of Integration**: Substitute the upper and lower bounds into the antiderivative: \( \ln|5| - \ln|1| \).
• **Simplification and Calculation**: Calculate the result: since \( \ln|1| = 0 \), the result simplifies to \( \ln 5 \).

The sequence of these steps provides a solid framework to solve any problem involving definite integration. It not only helps in gaining a numerical understanding of the region but also improves problem-solving skills in calculus.