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A definite integral is a fundamental concept in calculus which represents the accumulation of a quantity, such as area under a curve, over a specific interval \([a, b]\). It is denoted as \( \int_{a}^{b} f(x) \, dx \), where \(f(x)\) is a function defined on the interval. In practical terms, this integral gives the total amount of a function's output over the range from \(a\) to \(b\).
In our particular exercise, we are focusing on the integral of \( \int_{1}^{3} \left( \frac{1}{x} \right) dx \), which represents the area under the curve \( y = \frac{1}{x}\) between \(x=1\) and \(x=3\). This integral can be precisely evaluated using standard techniques or approximated using numerical methods such as Riemann sums.
Understanding how to compute the definite integral helps in solving various real-world problems, such as finding distances, areas, volumes, averages, and more.

Numerical integration is a method used to calculate the value of a definite integral when an exact form is either difficult or impossible to find. Instead of solving the integral analytically, which involves finding an antiderivative, numerical integration approximates the value using a series of computations.
There are several techniques within numerical integration, with Riemann sums being a common initial approach. By dividing the area under a curve into subintervals, different sums can be calculated, each providing an approximation of the integral's value. These sums include the left endpoint, right endpoint, and midpoint methods, each differing in how they choose points for evaluation:

• Left Endpoint Method: Uses the leftmost point of each subinterval.
• Right Endpoint Method: Uses the rightmost point of each subinterval.
• Midpoint Method: Uses the central point of each subinterval.

Numerical methods are invaluable in applications where precision is less critical, or an exact solution is too complex.

The left endpoint method is one of the strategies used to approximate the value of a definite integral via Riemann sums. It involves calculating the height of rectangles under a curve using the leftmost point of each subinterval.
To implement this, first determine the width of each subinterval as \( \Delta x = \frac{b-a}{n} \), where \(a\) and \(b\) are the boundaries of the interval, and \(n\) is the number of subintervals. For each rectangle, the height is set by evaluating the function at the left endpoint of the subinterval. The sum is then calculated as:
\[ L_n = \sum_{i=0}^{n-1} f(x_i) \cdot \Delta x \]
In this method, since the intervals are based on lower values, the left endpoint tends to give a less accurate approximation when the function is increasing, leading to an underestimate of the integral's true value.

This method is a simple variation of the left endpoint method used in Riemann sums. It evaluates the height of the rectangles using the rightmost endpoint of each subinterval, thus differing slightly in its approach.
As with the left endpoint method, calculate the width of each subinterval using \( \Delta x \). However, instead of the first subinterval starting at \(a\), the intervals use endpoints like \(x_1, x_2, \, \ldots, x_{n}\), with the sum expressed as:
\[ R_n = \sum_{i=1}^{n} f(x_i) \cdot \Delta x \]
This time, the method is based on using the rightmost side of the subinterval for evaluation. In cases where the function is increasing, the right endpoint method often results in an overestimate of the true integral value. This approach, just like the left endpoint method, is useful for gaining a quick understanding of the behavior of the integral over a given interval.

Among the Riemann sum approaches, the midpoint method often provides a more balanced and typically more accurate approximation for calculating a definite integral.
As its name suggests, this method uses the midpoint of each subinterval to determine the height of the rectangles. Begin as before by dividing the range into subintervals with width \( \Delta x \). The midpoint of each subinterval, say \(x_i^* \), is computed as \(a + \left(i + \frac{1}{2}\right) \cdot \Delta x\), and the sum is found by:
\[ M_n = \sum_{i=0}^{n-1} f(x_i^*) \cdot \Delta x \]
The midpoint method tends to be more accurate since it takes into account the function's value around the center of each subinterval, effectively balancing the potential overestimation by a rising function and underestimation by a descending one.