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Approximating Area and Riemann Sums

Objectives

  • Use sigma (summation) notation to calculate sums and powers of integers.
  • Use the sum of rectangular areas to approximate the area under a curve.
  • Use Riemann sums to approximate area.

Summary

A Riemann sum is simply a sum of products of the form f ( x i ∗ ) Δ x that estimates the area between a positive function and the horizontal axis over a given interval. If the function is sometimes negative on the interval, the Riemann sum estimates the difference between the areas that lie above the horizontal axis and those that lie below the axis.

The three most common types of Riemann sums are left, right, and middle sums, but we can also work with a more general Riemann sum. The only difference among these sums is the location of the point at which the function is evaluated to determine the height of the rectangle whose area is being computed. For a left Riemann sum, we evaluate the function at the left endpoint of each subinterval, while for right and middle sums, we use right endpoints and midpoints, respectively.

Formulas

The left, right, and middle Riemann sums are denoted L n , R n , and M n , with formulas

L n = f ( x 0 ) Δ x + f ( x 1 ) Δ x + ⋯ + f ( x n − 1 ) Δ x = ∑ i = 0 n − 1 f ( x i ) Δ x , R n = f ( x 1 ) Δ x + f ( x 2 ) Δ x + ⋯ + f ( x n ) Δ x = ∑ i = 1 n f ( x i ) Δ x , M n = f ( x ¯ 1 ) Δ x + f ( x ¯ 2 ) Δ x + ⋯ + f ( x ¯ n ) Δ x = ∑ i = 1 n f ( x ¯ i ) Δ x ,

where x 0 = a , x i = a + i Δ x , and x n = b , using Δ x = b − a n . For the midpoint sum, x ¯ i = ( x i − 1 + x i ) / 2 .

See the Desmos Demonstration.

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