skeeter said:
karush said:
A Riemann sum is a method used in calculus to approximate the area under a curve by dividing it into smaller rectangles and adding up their areas. It is an important concept in AP Calculus as it is used to find the exact value of integrals.
To calculate a Riemann sum, the interval of the curve is divided into n smaller intervals, and the width of each interval is determined. Then, the function is evaluated at a specific point within each interval, and the product of the width and the function value is added up for all the intervals to find the approximate area under the curve.
Riemann sum is significant in AP Calculus as it is used to approximate the area under a curve and find the exact value of integrals. It is also used to determine the convergence or divergence of infinite series and to calculate the volume of irregularly shaped objects.
The three types of Riemann sums are left Riemann sum, right Riemann sum, and midpoint Riemann sum. In left Riemann sum, the function is evaluated at the left endpoint of each interval, while in right Riemann sum, the function is evaluated at the right endpoint of each interval. In midpoint Riemann sum, the function is evaluated at the midpoint of each interval.
Riemann sum is closely related to the definite integral as the definite integral is defined as the limit of Riemann sums as the number of intervals approaches infinity. In other words, the definite integral is the exact value of the area under a curve, while Riemann sum is an approximation of this value.
