A Simple Riemann Sum is a method used in calculus to approximate the area under a curve on a graph. It involves dividing the area into smaller rectangles and summing their areas to get an estimate of the total area.
To calculate a Simple Riemann Sum, you first divide the interval of the curve into smaller sub-intervals. Then, you choose a sample point within each sub-interval and calculate the area of the rectangle formed by that sample point and the height of the curve at that point. Finally, you add up the areas of all the rectangles to get the approximate area under the curve.
The purpose of using Simple Riemann Sums is to approximate the area under a curve when it is not possible to find the exact area using traditional methods. It is also useful for understanding the concept of integration in calculus.
Some limitations of Simple Riemann Sums include its reliance on choosing appropriate sub-intervals and sample points, which can affect the accuracy of the approximation. It also only gives an estimate of the area and not the exact value.
Simple Riemann Sums differ from other methods such as the Trapezoidal Rule and Simpson's Rule in that it uses rectangles to approximate the area, whereas the other methods use trapezoids and parabolas, respectively. Simple Riemann Sums also only use a single sample point within each sub-interval, while the other methods use multiple points to improve accuracy.