A Riemann Sum is a process used for approximating the value of a definite integral using subintervals. The subintervals do not have to be of equal length, and as the number of subintervals increases, so does the accuracy of the Riemann Sum. There are several types of Riemann Sums: left, right, midpoint, and trapezoidal.

The basic concept behind Riemann Sums is to create a rectangle from each subinterval, then find the total area of all the rectangles to calculate an approximation for the area under the curve of the function. The widths of the rectangles are the widths of the subintervals, while the heights of the rectangles are determined by the function. The method for finding the height of the subinterval rectangles is different for each of the types of Riemann Sums. Left, right, and midpoint Riemann Sums are described below.

Types of Riemann Sums

Left Riemann Sum

In a Left Riemann Sum, the height of each subinterval is determined by the value of the function at the left endpoint of that subinterval. As shown by the graph to the right, the first subinterval, from 0
Left Riemann Sum
Left Riemann Sum
to 0.5, has a height of 0 because the function (y=x³) at the left endpoint has a value of 0. The second subinterval, from 0.5 to 1, has a height of 0.125 because the function at x=0.5 has a value of 0.125. This continues along the function until you have subintervals along the entire length you want to approximate the area of.

To calculate the area under the function y=x³ using a Left Riemann Sum approximation and 4 equal subintervals, on the interval x=0 to x=2, take the following steps.
  • Graph the function along the interval (0 to 2)
  • Create points on the x-axis so that you have four equal subintervals. For this example, in addition to the two boundaries x=0 and x=2, also include x=0.5, 1, and 1.5.
  • Determine the height of each of the four subinterval rectangles by finding the value of the function at the x-value that is on the left side of the rectangle. Draw a line up from the x-axis to the function, then over to the right side of that rectangle. Do this for each of the four rectangles.
  • Calculate the area of each of the rectangles by multiplying the height by the width. The width of the subintervals is found by finding the distance along the total interval (upper bound minus lower bound), then dividing by the total number of subintervals. In this case, the width is 0.5 because (2-0)/4=0.5.
    • Rectangle 1: height=0, width=0.5, area=0x0.5=0
    • Rectangle 2: height=0.125, width=0.5, area=0.125x0.5=0.0625
    • Rectangle 3: height=1, width=0.5, area=1x0.5=0.5
    • Rectangle 4: height=3.375, width=0.5, area=3.375x0.5=1.6875
  • Add the areas of all the rectangles together to calculate the total area under the function to find the approximation using the Left Riemann Sum approximation.
    • Total area: 0 + 0.0625 + 0.5 + 1.6875=2.25

Right Riemann Sum

In a Right Riemann Sum, the height of each subinterval is determined by the value of the function at the right endpoint of that subinterval. As shown by the
Right Riemann Sum
Right Riemann Sum
graph to the right, the first subinterval, from 0 to 0.5, has a height of 0.125 because the function (y=x³) at the right endpoint has a value of 0.125. The second subinterval, from 0.5 to 1, has a height of 1 because the function at x=1 has a value of 1. This continues along the function until you have subintervals along the entire length you want to approximate the area of.

To calculate the area under the function y=x³ using a Left Riemann Sum approximation and 4 equal subintervals, on the interval x=0 to x=2, take the following steps.
  • Graph the function along the interval (0 to 2)
  • Create points on the x-axis so that you have four equal subintervals. For this example, in addition to the two boundaries x=0 and x=2, also include x=0.5, 1, and 1.5.
  • Determine the height of each of the four subinterval rectangles by finding the value of the function at the x-value that is on the right side of the rectangle. Draw a line up from the x-axis to the function, then over to the left side of that rectangle. Do this for each of the four rectangles.
  • Calculate the area of each of the rectangles by multiplying the height by the width. The width of the subintervals is found by finding the distance along the total interval (upper bound minus lower bound), then dividing by the total number of subintervals. In this case, the width is 0.5 because (2-0)/4=0.5.
    • Rectangle 1: height=0.125, width=0.5, area=0.125x0.5=0.0625
    • Rectangle 2: height=1, width=0.5, area=1x0.5=0.5
    • Rectangle 3: height=3.375, width=0.5, area=3.375x0.5=1.6875
    • Rectangle 4: height=8, width=0.5, area=8x0.5=4
  • Add the areas of all the rectangles together to calculate the total area under the function to find the approximation using the Right Riemann Sum approximation.
    • Total area: 0.0625 + 0.5 + 1.6875 + 4=6.25

Midpoint Riemann Sum

external image MidRiemann2.PNGIn a Midpoint Riemann Sum, the height of each subinterval is determined by the value of the function at the center of that subinterval. As shown by the graph to the right, the first subinterval, from 0 to 0.5, has a height of 0.015625 because the function (y=x³) at x=0.25 (halfway between the two endpoints of the subinterval) has a value of 0.015625. The second subinterval, from 0.5 to 1, has a height of 0.421875 because the function at x=0.75 has a value of 0.421875. This continues along the function until you have subintervals along the entire length you want to approximate the area of.

To calculate the area under the function y=x³ using a Midpoint Riemann Sum approximation and 4 equal subintervals, on the interval x=0 to x=2, take the following steps.
  • Graph the function along the interval (0 to 2)
  • Create points on the x-axis so that you have four equal subintervals. For this example, in addition to the two boundaries x=0 and x=2, also include x=0.5, 1, and 1.5.
  • Determine the height of each of the four subinterval rectangles by finding the value of the function at the x-value that is in the center of each subinterval (center value is equal to the average of the two endpoints). Draw a line up from each endpoint on the subinterval to the y-value of the midpoint, then connect them with a horizontal line that passes through the graph at the midpoint. Do this for each of the four rectangles.
  • Calculate the area of each of the rectangles by multiplying the height by the width. The width of the subintervals is found by finding the distance along the total interval (upper bound minus lower bound), then dividing by the total number of subintervals. In this case, the width is 0.5 because (2-0)/4=0.5.
    • Rectangle 1: height=0.125625, width=0.5, area=0.125625x0.5=0.0078125
    • Rectangle 2: height=0.421875, width=0.5, area=0.421875x0.5=0.2109375
    • Rectangle 3: height=1.953125, width=0.5, area=1.953125x0.5=0.9765625
    • Rectangle 4: height=5.359375, width=0.5, area=5.359375x0.5=2.6796875
  • Add the areas of all the rectangles together to calculate the total area under the function to find the approximation using the Midpoint Riemann Sum approximation.
    • Total area: 0.0078125 + 0.2109375 + 0.9765625 + 2.6796875=3.875

Find an example of a Right Riemann Sum here.

Other Notes

Midpoint Riemann Sums usually give the most accurate approximation of the three methods.
It should be noted that when calculating Left and Right Riemann Sums, one will yield what is called an "Upper Sum" and the other will yield a "Lower Sum." The Upper Sum is the approximation that is the greater of the two, while the Lower Sum is the lesser. The Upper Sum, Lower Sum, and the actual area under the curve of the function share the following relationship:
  • Lower Sum ≤ Actual Area ≤ Upper Sum

Sources:

Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards. Calculus. 8th ed. Boston and New York: Houghton Mifflin Company, 2006. Pages: 259-281.

"Riemann Sum." Wikipedia: the Free Encyclopedia. 14 May 2007. < http://en.wikipedia.org/wiki/Riemann_sums >.