Difficulty: Intermediate | Prerequisites: MAT210 Section 4.1 (Antidifferentiation), basic geometry (area of rectangles and triangles)
This topic sits at the start of the integration unit in MAT210. Up to now you have been doing derivatives (rates of change) and have just learned antidifferentiation, which reverses that process. Here, the course asks a new question: if you know how fast something is changing, can you find the total amount that has accumulated? The answer turns out to be the area under a curve, and the tool for computing it is the definite integral. Understanding this link between area, accumulation, and antiderivatives is the foundation for everything that follows in the integration chapters.
When you graph a rate function, the area between the curve and the horizontal axis equals the total accumulated quantity. For simple shapes (rectangles, triangles) you can compute this area with geometry. For curves where no neat geometric formula exists, you approximate the area with rectangles, a technique called Riemann summation, and the exact answer is written as a definite integral.
Rate function
A function that describes how quickly a quantity changes per unit of time (or another variable). Written with a prime, e.g. R'(t) or f'(t).
Think of it as the speedometer reading: it tells you how fast, not how far.
Accumulation
The total quantity that builds up over an interval. Found by multiplying rate by time (for a constant rate) or by computing the area under the rate function's graph.
In simple terms, this is the "how much in total" answer you get from a "how fast" function.
Antiderivative
A function F(x) whose derivative is f(x). If f'(t) = 1200, then F(t) = 1200t is an antiderivative because differentiating 1200t gives back 1200.
Think of it as running the derivative process in reverse.
Definite integral
The notation for the exact accumulated quantity over an interval [a, b]. Written as the integral from a to b of f(x) dx. It represents the exact area between the graph of f(x) and the x-axis from x = a to x = b.
In simple terms, this is the formal way of writing "total area under the curve between two points."
Riemann summation
A method of approximating the area under a curve by dividing the region into narrow rectangles, computing each rectangle's area, and adding them up.
Think of it as filling the space under a curve with thin bars and counting up the total.
When the rate of change is constant, the graph of the rate function is a horizontal line. The accumulated total equals the area of the rectangle formed under that line.
Parking garage example. A garage earns $1,200 per day. The rate function is R'(t) = 1200, where t is in days.
After 3 days: (1200 $/day)(3 days) = $3,600
After 7 days: (1200 $/day)(7 days) = $8,400
After 15 days: (1200 $/day)(15 days) = $18,000
Each of these calculations is the area of a rectangle: width = number of days, height = rate per day.
The general accumulation function is R(t) = 1200t. Two things follow from this:
R'(t) = 1200 is the derivative of R(t) = 1200t.
R(t) = 1200t is an antiderivative of R'(t) = 1200.
So the accumulated quantity (area under the rate curve) turns out to be the antiderivative of the rate function. This is the central insight of the section.
When the rate function is a line through the origin (not horizontal), the area under the graph is a triangle rather than a rectangle.
Arena example. An arena opens at 5 p.m. for a concert. The rate at which people enter is f'(t) = 2t, where t is minutes after 5 p.m. and f'(t) is people per minute.
After 60 minutes the graph of f'(t) = 2t from t = 0 to t = 60 forms a triangle:
Base = 60 minutes
Height = f'(60) = 120 people/min
Area = (1/2)(60)(120) = 3,600 people
For a general time t:
Base = t, height = 2t
Area = (1/2)(t)(2t) = t squared
So f(t) = t squared is the accumulation function, and it is an antiderivative of f'(t) = 2t. You can verify this with the power rule: the integral of 2t dt = (2t squared)/2 = t squared + C.
The same pattern holds: the area under the rate curve equals the antiderivative of the rate function.
The definite integral is the formal notation for the accumulated total over an interval. It is written as the integral from a to b of f(x) dx and represents the area between the graph of f(x) and the x-axis over the interval [a, b].
Parking garage in integral notation:
Integral from 0 to 3 of 1200 dt = $3,600 (total revenue, first 3 days)
Integral from 0 to 7 of 1200 dt = $8,400 (total revenue, first 7 days)
Integral from 0 to 15 of 1200 dt = $18,000 (total revenue, first 15 days)
Arena in integral notation:
Integral from 0 to 60 of 2t dt = 3,600 (total people in the arena after 60 minutes)
The integral sign is an elongated S (for "sum"), the limits a and b mark the start and end of the interval, f(x) is the function being integrated, and dx names the variable. These four components appear in every definite integral.
Area of a rectangle
Area = base x height. Used when the rate function is constant (horizontal line).
Area of a triangle
Area = (1/2) x base x height. Used when the rate function is a line through the origin.
Riemann sum (left or right)
Sum from i = 1 to n of f(t_i) x delta-t, where delta-t is the width of each subinterval.
Definite integral (exact area)
The limit of the Riemann sum as delta-t approaches 0 equals the integral from a to b of f(x) dx.
Note that delta-t becomes dt and the summation sign becomes the integral sign in the transition from approximation to exact value.
This technique is how businesses calculate total revenue or profit from a known rate of earning. If you know the marginal profit each day (the rate), the definite integral gives you the total profit over a period.
The same logic applies in physics: if you know velocity (a rate of change of position), the area under the velocity-time graph gives the total distance travelled. Epidemiologists use it to find total infections from an infection-rate curve. In each case the pattern is identical: rate in, accumulated total out.
Students often confuse the rate function with the accumulation function. R'(t) = 1200 is the rate (dollars per day); R(t) = 1200t is the total accumulated. The rate tells you how fast; the accumulation tells you how much.
Students sometimes plug the rate function's value directly in as the total. For example, f'(60) = 120 tells you 120 people per minute are entering at minute 60, not that 120 people have entered in total. The total requires computing the area.
Left-rectangle and right-rectangle Riemann sums are not the same. For an increasing function, the left sum underestimates and the right sum overestimates. Students often forget which is which.
The area under the curve has units. If the y-axis is dollars/day and the x-axis is days, the area is in dollars. Forgetting to interpret the units leads to nonsensical answers.
⚠️ You will likely be asked to compute a definite integral by identifying the geometric shape under the curve (rectangle or triangle) and using the appropriate area formula.
⚠️ Expect a Riemann sum problem: given a table of values, compute the left sum, right sum, or their average. Know which one overestimates and which underestimates for increasing vs decreasing functions.
⚠️ Be ready to interpret the result in context. "The integral from 0 to 10 of P'(t) dt = 368" means Acme, Inc. earned approximately $368,000 in total profit over 10 days, not that their daily profit is $368,000.
⚠️ The connection between the antiderivative and the area under the curve is the bridge to the Fundamental Theorem of Calculus, which is coming in the next lesson. This section sets up that theorem.
True or false: If R'(t) = 1200, the total revenue after 5 days is $6,000. (True.)
Fill in the blank: The area under a constant rate function over an interval forms a ________. (Rectangle.)
True or false: For an increasing function, the left Riemann sum overestimates the area. (False. It underestimates.)
Fill in the blank: The accumulation function is an ________ of the rate function. (Antiderivative.)
True or false: The definite integral from 0 to 60 of 2t dt equals 3,600. (True.)
Q: A machine produces widgets at a constant rate of 50 widgets per hour. Using a definite integral, express and compute the total number of widgets produced in 8 hours.
A: Integral from 0 to 8 of 50 dt = 50 x 8 = 400 widgets. The area is a rectangle with height 50 and width 8.
Q: The rate at which water flows into a tank is given by f'(t) = 4t litres per minute. How many litres have entered the tank after 10 minutes?
A: The graph of f'(t) = 4t from 0 to 10 is a triangle with base 10 and height 40. Area = (1/2)(10)(40) = 200 litres.
Q: A company's marginal profit is given by P'(t) = 0.8t squared + 10 (thousands of dollars per day). Using a left Riemann sum with delta-t = 1 over [0, 5], approximate the total profit.
A: Heights at t = 0, 1, 2, 3, 4: P'(0) = 10, P'(1) = 10.8, P'(2) = 13.2, P'(3) = 17.2, P'(4) = 22.8. Sum = 10 + 10.8 + 13.2 + 17.2 + 22.8 = 74. Total profit is approximately $74,000.
Q: Explain why the left Riemann sum underestimates the area for an increasing function.
A: For an increasing function, the left endpoint of each subinterval is the lowest point. The rectangle's height is therefore shorter than the curve over most of the subinterval, leaving a gap between the top of the rectangle and the curve. That gap is area the sum misses.
This material connects directly to MAT210 Section 4.1 (Antidifferentiation) because the accumulation function is the antiderivative of the rate function. That link is formalised in the next lesson as the Fundamental Theorem of Calculus, which ties the definite integral to the antiderivative and eliminates the need for Riemann sums when an antiderivative can be found.
The concept of accumulation from a rate also connects to the application chapters on consumer surplus, producer surplus, and present value of income streams, all of which rely on definite integrals.
Integration, definite integral, area under a curve, accumulation, antiderivative, antidifferentiation, rate function, Riemann sum, Riemann summation, left rectangles, right rectangles, constant rate, linear rate, MAT210, Brief Calculus, ASU, Section 4.2, power rule of antidifferentiation, total revenue, marginal profit, rectangle area, triangle area, integral notation, geometric area, area approximation