Riemann Sum Problems

Problems that require students to determine left, right, midpoint, trapezoidal, upper or lower Riemann sums are frequent in AP Calculus AB tests.

Consider a function f defined on a subset of the real numbers, and let I = [a, b] be a closed interval contained in the subset. A finite set of points {x₀, x₁, x₂, ... x_n} with a = x₀ < x₁ < x₂ ... < x_n = b creates a partition {[x₀, x₁), [x₁, x₂), ... [x_n-1, x_n]} of the closed interval I. The Riemann sum of f over interval I is:


Left Riemann sum: y_i = x_i-1
Right Riemann sum: y_i = x_i
Midpoint Riemann sum: y_i = (x_i + x_i-1)/2
Trapezoidal Riemann sum: average of the left and right Riemann sums
Upper Riemann sum: y_i = supremum of f over [x_i-1, x_i]
Lower Riemann sum: y_i = infimum of f over [x_i-1, x_i]

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Examples of problems with Riemann sums

Question 1:

t(hours)	0	.5	1	1.5	2	2.5	3
v(miles/hour)	32	30	16	22	20	24	26

A speedboat travels downstream on a river. Its speed v, in miles per hour, at certain times is given in the table above. Using a left Riemann sum, what is the approximation of the total distance traveled by the speedboat from t = .5 to t = 3?

Solution: The left Riemann sum is:

d = 30(1-.5) + 16(1.5-1) + 22(2-1.5) + 20(2.5-2) + 24(3-2.5).
d = 56 miles

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Question 2:

The values of a differentiable function f are given in the table below. Aproximate the integral of this function from x = 0 to x = 8, by using a right Riemann sum.

x	0	2	4	6	8
f(x)	-10	-2	6	8	14

Solution: The left Riemann sum is:
R = -2(2-0) + 6(4-2) + 8(6-4) + 14(8-6).
R = 52

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