Reemann sum

A Riemann sum is a method used to approximate the area under a curve. It involves dividing the interval over which the area is to be calculated into smaller sub-intervals, and approximating the area in each sub-interval using rectangles.

To calculate a Riemann sum, first, the interval [a, b] is divided into n sub-intervals, each of width $\Delta x = \frac{b - a}{n}$. Then, a representative point, xi, is chosen in each sub-interval. The area of each rectangle is calculated as f(xi) * $\Delta x$, where f(xi) represents the value of the function at the chosen point xi. Finally, all the areas of the rectangles are added together to obtain an approximation of the total area under the curve.

There are different methods for choosing the representative points within each sub-interval, such as the left endpoint, right endpoint, or midpoint of the sub-interval. The choice of the method can affect the accuracy of the approximation.

As the number of sub-intervals, n, approaches infinity, the Riemann sum becomes a definite integral, and the approximation becomes more accurate.