Difficulty: Intermediate | Prerequisites: MAT210 Section 4.1 (Antidifferentiation), Section 4.2 Part A/B (Area under a curve for constant and linear rate functions)
The earlier parts of Section 4.2 showed that the area under a rate function equals the accumulated total, and that you can compute this area with basic geometry when the rate function is a horizontal line or a straight line through the origin. Most real-world rate functions are curves, though, and no simple geometric formula covers a parabola or an exponential. Riemann summation is the workaround: approximate the curved region with narrow rectangles, add up their areas, and get as close to the true total as you need. This technique is what makes the definite integral possible for any continuous function, and it is the conceptual bridge between geometry and the formal definition of the integral.
When the rate function is a curve (not a straight line), you approximate the area under it by slicing the interval into rectangles, computing each rectangle's area, and adding them. Using the left endpoints gives a left Riemann sum; using the right endpoints gives a right Riemann sum. As the rectangles get narrower and more numerous, the sum converges to the exact value, which is the definite integral.
Riemann sum
An approximation of the area under a curve, found by dividing the interval into n subintervals, drawing a rectangle on each, and adding their areas. Named after Bernhard Riemann.
Think of it as filling the space under a curve with thin bars and totalling up the bar areas.
Left Riemann sum
A Riemann sum where each rectangle's height is determined by the function value at the left endpoint of its subinterval.
For an increasing function, this underestimates the true area because the rectangle is shorter than the curve over most of the subinterval.
Right Riemann sum
A Riemann sum where each rectangle's height is determined by the function value at the right endpoint of its subinterval.
For an increasing function, this overestimates the true area because the rectangle is taller than the curve over most of the subinterval.
Subinterval
One of the n equal slices you divide the interval [a, b] into. Each subinterval has width delta-t = (b - a) / n.
In simple terms, it is one "slice" of the x-axis that gets its own rectangle.
Delta-t (or delta-x)
The width of each subinterval (and each rectangle). As delta-t shrinks toward zero, the Riemann sum approaches the definite integral.
Think of it as how thin you cut each slice. Thinner slices give a more accurate total.
A constant rate function graphs as a horizontal line, giving a rectangle. A linear rate function through the origin gives a triangle. Both have standard area formulas.
A quadratic rate function, such as P'(t) = 0.8t squared + 10, graphs as a parabola. There is no elementary geometric formula for the area under a parabola. The same is true for exponentials, logarithms, and most other non-linear functions you will meet in calculus.
The solution is to approximate the curved region with shapes we can measure: rectangles. This is Riemann summation.
Setup. The rate of change of profit is P'(t) = 0.8t squared + 10 (thousands of dollars per day), and we want the total profit over [0, 10]. We use 10 subintervals, each of width delta-t = 1.
First, build the table of values:
t | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|
P'(t) | 10 | 10.8 | 13.2 | 17.2 | 22.8 | 30 | 38.8 | 49.2 | 61.2 | 74.8 | 90 |
Left Riemann sum. Use all points except the rightmost (t = 10). Each rectangle has base 1 and height P'(t_i) at the left endpoint.
Left sum = 10 + 10.8 + 13.2 + 17.2 + 22.8 + 30 + 38.8 + 49.2 + 61.2 + 74.8 = 328
Because P'(t) is increasing on this interval, the left sum underestimates the true area.
Right Riemann sum. Use all points except the leftmost (t = 0). Each rectangle has base 1 and height P'(t_i) at the right endpoint.
Right sum = 10.8 + 13.2 + 17.2 + 22.8 + 30 + 38.8 + 49.2 + 61.2 + 74.8 + 90 = 408
Because P'(t) is increasing, the right sum overestimates the true area.
Average of the two sums. (328 + 408) / 2 = 368
So the integral from 0 to 10 of (0.8t squared + 10) dt is approximately 368, meaning Acme, Inc. earned roughly $368,000 in total profit over the first ten days.
The Riemann sum with 10 rectangles gives an approximation. If you use 100 rectangles (delta-t = 0.1), the approximation improves. With 1,000 rectangles, it improves further.
The definite integral is defined as the limit of this process. As delta-t approaches zero (and the number of rectangles n approaches infinity), the Riemann sum converges to the exact area:
limit as delta-t approaches 0 of the sum from i = 1 to n of P'(t_i) times delta-t = the integral from 0 to 10 of P'(t) dt
Two notational shifts happen in this limit:
The summation sign (capital sigma) becomes the integral sign (elongated S).
The finite width delta-t becomes the infinitesimal dt.
This is the formal definition of the definite integral. The next lesson (the Fundamental Theorem of Calculus) will provide a shortcut so you do not have to compute this limit every time: you evaluate the antiderivative at the endpoints instead.
Students sometimes think the left and right sums always give the same answer. They do not. They only coincide when the function is constant. For any other function, one will be larger than the other.
Students often assume the average of the left and right sums is the exact answer. It is a better approximation than either sum alone, but it is still an approximation. The exact answer requires taking the limit (or using the Fundamental Theorem of Calculus).
Students sometimes forget that the height of each rectangle comes from evaluating the function, not from reading the t-value itself. The height is P'(t_i), not t_i.
Students sometimes confuse "more rectangles" with "larger area." More rectangles (smaller delta-t) give a more accurate approximation, not a bigger one. The area under the curve is fixed; you are just measuring it more precisely.
⚠️ You will likely be given a table of function values and asked to compute a left sum, a right sum, or both, then average them. Know the procedure cold.
⚠️ Be prepared to state whether a given Riemann sum is an overestimate or an underestimate, and explain why based on whether the function is increasing or decreasing.
⚠️ Understand the notation: the summation from i = 1 to n of f(t_i) times delta-t is the Riemann sum, and its limit as delta-t approaches 0 is the definite integral. Expect to see both forms and be asked to explain the connection.
⚠️ The exact value of the Acme profit integral (using antidifferentiation) is 366.67, confirming that 368 from the Riemann average is indeed close. You may be asked to compare.
True or false: A left Riemann sum always underestimates the area under a curve. (False. It underestimates for increasing functions but overestimates for decreasing functions.)
Fill in the blank: As the number of rectangles increases, the Riemann sum gets closer to the ________. (Definite integral.)
True or false: The average of the left and right Riemann sums is the exact area. (False. It is a better approximation, but still an approximation.)
Fill in the blank: In the limit, the summation sign becomes the ________ sign. (Integral.)
Q: Given P'(t) = 0.8t squared + 10, compute the right Riemann sum over [0, 5] with delta-t = 1.
A: Heights at t = 1, 2, 3, 4, 5: 10.8, 13.2, 17.2, 22.8, 30. Right sum = 10.8 + 13.2 + 17.2 + 22.8 + 30 = 94. Total profit is approximately $94,000 over the first five days.
Q: Is the right sum from the previous question an overestimate or an underestimate? Explain.
A: It is an overestimate because P'(t) = 0.8t squared + 10 is increasing on [0, 5]. The right endpoint of each subinterval is the highest point, so each rectangle sticks up above the curve.
Q: If you wanted a more accurate Riemann sum for the Acme problem, what would you change?
A: Use more rectangles (smaller delta-t). For example, 100 rectangles with delta-t = 0.1 instead of 10 rectangles with delta-t = 1. The approximation converges to the exact integral as delta-t approaches 0.
Q: Write the formal definition of the definite integral from a to b of f(t) dt in terms of a Riemann sum and a limit.
A: The integral from a to b of f(t) dt = limit as delta-t approaches 0 of the sum from i = 1 to n of f(t_i) times delta-t, where delta-t = (b - a) / n and t_i is a point in the i-th subinterval.
Riemann summation is the conceptual engine behind the definite integral, and the next lesson (the Fundamental Theorem of Calculus) will show that you can skip the rectangle-and-limit process entirely by evaluating an antiderivative at the endpoints of the interval.
The Riemann sum idea reappears in numerical methods courses and in applied settings where the function has no closed-form antiderivative (you approximate the integral numerically instead). It also connects to sigma notation and sequences, topics from precalculus that are worth revisiting if the summation symbol feels unfamiliar.
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