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In numerical integration, the Left Endpoint Approximation is a simple method to estimate the area under a curve. Think of it like drawing rectangles under the graph of a function, where each rectangle's left side touches the curve. This touchpoint is the height of the rectangle. To find the approximation, we sum the areas of these rectangles. The formula used is:

• \[A_L = \sum_{i=0}^{n-1} f(x_i) \cdot \Delta x\]

Here, \(n\) is the number of subintervals you divide your full interval into, and \(\Delta x = \frac{b-a}{n}\) is the width of each rectangle.
By using the left endpoint \(x_i = 1 + i \cdot \Delta x\), where \(a=1\) and \(b=2\), you calculate the value at this point with your function \(f(x) = \frac{1}{x}\). So, the left endpoint approximation captures an area with rectangles that lean left. Note that this approach might underestimate areas when your curve is increasing, but it gives a helpful first glance at approximating the area.

The Midpoint Approximation is another approach in numerical integration, often more accurate than the left endpoint approximation. Here, each rectangle's height is determined by the value of the function at the midpoint of its base. This creates a more balanced and potentially more accurate estimation.To compute this, the formula is:

• \[A_M = \sum_{i=0}^{n-1} f\left(x_i + \frac{\Delta x}{2}\right) \cdot \Delta x\]

Each rectangle's base is still \(\Delta x\) wide, but to get the height, we evaluate the function at the midpoint \(x_i + \frac{\Delta x}{2}\).
For example, if \(x_i\) are the left endpoints of each subinterval \([1,2]\), and considering subintervals like \(n=10\), you add half of \(\Delta x\) to find the center or middle point. This mid-value better captures the average height under the curve, serving as a good balance and often resulting in a closer approximation to the true integral.

The Right Endpoint Approximation helps us estimate the area under a curve by using rectangles that touch the curve at their rightmost side. This technique often overestimates the area if the function is decreasing, but it's handy for gaining insights about the function's behavior.The formula works as follows:

• \[A_R = \sum_{i=1}^{n} f(x_i) \cdot \Delta x\]

Here, \(x_i = 1 + i \cdot \Delta x\) for the right endpoint is chosen, where \(\Delta x\), as before, identifies the width.
Instead of starting, as we did in the left endpoint, at \(x=1\), we now look at the interval from the left of each subinterval (since we're calculating with respect to the right endpoint), so \(i = 1\) to \(n\). For instance, when dividing into \(n=10\) parts, you would get heights based on right-edge points like \(x=1.1, 1.2, ...\), leading to a fresh perspective that fills the remaining unseen gaps when using left approximations. Together with midpoint and left endpoint estimates, right endpoint adds depth to numerical integration techniques.