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Approximating the area under a curve using rectangles, as demonstrated in this exercise, is a classic approach in calculus known as the Riemann sum. This method allows us to estimate the integral of a function over an interval by dividing the interval into smaller sub-intervals. The choice of how you draw rectangles—such as using left endpoints, right endpoints, or midpoints—determines the type of Riemann sum you achieve.

In our problem, we employed the left-endpoint approximation method. This involves evaluating the function at the left endpoint of each rectangle's base to determine its height. Here’s how it works:

• Divide the interval into equal parts.
• Calculate the width of each rectangle (\(\Delta x = \frac{b-a}{n}\)).
• Determine the height of the rectangles by evaluating the function at the left endpoints \(f(x_i)\).

This method provides a clearer image of how the function behaves across an interval by making small approximations, which get more accurate as the number of rectangles increases.

The definite integral plays a crucial role in understanding the area under a curve. Mathematically, it provides the exact value of the area for a continuous function over a specified interval. In this case, we are dealing with the integral of the logarithmic function \(f(x) = \ln x\) over the interval \([1, 2]\).

The integral is represented as \(\int_{1}^{2} \ln x \, dx\). Calculating this offers the exact area under the curve between the specified limits. Here’s a simple breakdown:

• The definite integral sums up small "slices" or areas across an interval to give us a total area.
• Though challenging to compute manually for complex functions, numerical techniques like Riemann sums approximate it well.

The approximation becomes significantly accurate with increasing rectangles, giving us insight into the behavior and characteristics of the function over its interval.

The logarithmic function \(f(x) = \ln x\) is a vital concept in mathematics, notable for its properties and application in various fields like biology, economics, and computer science. A logarithmic function is the inverse of an exponential function and has unique characteristics.

Key features include:

• It's only defined for positive values of \(x\) because logarithms of non-positive numbers are undefined.
• The function \(\ln x\) is continuous and differentiable for \(x > 0\).
• Graphically, it passes through the point \((1,0)\) and approaches infinity as \(x\) increases while descending to \(-\infty\) as \(x\) approaches zero positive from the right.

Understanding \(\ln x\) is essential when working on problems involving exponential decay, growth, and in this particular case, integrating over a specific interval. It helps give depth to how such functions behave and interact with limits and integrals.