Riemann sum

Overview

An approximation of the area under a curve by dividing it into simple shapes, and summing those composite areas .

where , the width of each shape, usually rectangles.

From definite integral to Riemann sum

From Riemann sum to definite integral

Examples

Detailed example

We can convert between definite integral and Riemann sum (where is the number of rectangles):

Turn the original sum () to this form (), by changing the sign of :

From this equation, we can identify the number of rectangles for the Riemann sum is . Similarly, we extract the width of each rectangle:

Using the width of each rectangle to find endpoints:

We also extract the height of each rectangle:

Rewrite the function in terms of and , using the equation :

From here, we can deduce that and . Thus . We define the definite integral as:

Left Riemann sum

Divide the area into equal-width rectangles, where the height of each rectangle matches the function value at the left endpoint of its base.

Right Riemann sum

The height of each rectangle is equal to the value of the function at the right endpoint of its base.

Right Riemann sum

Approximate the area between and the -axis on the interval using Right Riemann sum with 9 equal subdivisions

then each rectangle has the height of and area of

So a Right Riemann sum is calculated as:

Midpoint Riemann sum

Divide the area is equal to the value of the function at the midpoint of its base

Trapezoidal Rule

Use equal-height trapezoids, each touches the curve at both of its top vertices

Example

Trapezoidal Sums

Approximate the area between the -axis and , from  to using a trapezoidal sum with 4 equal subdivisions.

Repeating with all 4 trapezoids:

Sage Code

# Sage
var('x, i')
def left_sum(func, low_bound, upper_bound, num_recs):
    w = (upper_bound - low_bound) / num_recs
    left_sum = sum(w * func(x=low_bound + i*w), i, 0, num_recs-1)
    return left_sum
 
def mid_sum(func, low_bound, upper_bound, num_recs):
    w = (upper_bound - low_bound) / num_recs
    mid_sum = sum(w * func(x=low_bound + (i+1/2)*w), i, 0, num_recs-1)
    return mid_sum
 
def right_sum(func, low_bound, upper_bound, num_recs):
    w = (upper_bound - low_bound) / num_recs
    right_sum = sum(w * func(x=low_bound + i*w), i, 1, num_recs)
    return right_sum
 
def trapezoid_sum(func, low_bound, upper_bound, num_recs):
    w = (upper_bound - lowlow_bound + (i+1)*w))/2, i, 0, num_recs-1)
    return trap_sum