Exploring accumulation of change
Definite integrals are interpreted as the accumulation of quantities. Learn why this is so and how this can be used to analyze real-world contexts.
The definite integral can be used to express information about accumulation and net change in applied contexts. Let's see how it's done.
Thinking about accumulation in a real world context
Say a tank is being filled with water at a constant rate of (liters per minute) for . We can find the volume of the water (in ) by multiplying the time and the rate:
Now consider this case graphically. The rate can be represented by the constant function :
Function r sub 1 is graphed. Time in minutes is on the x-axis, from 0 to 10. Rate, in liters per minute, is on the y-axis. The graph is a line. The line starts at (0, 5), extends horizontally to the right, and ends at (10, 5).
Each horizontal unit in this graph is measured in minutes and each vertical unit is measured in liters per minute, so the area of each square unit is measured in liters:
A square represents a unit on a graph. The horizontal width represents minutes and the vertical height represents liters per minute. The area within represents liters. The equation to calculate area is width times height = area, or minutes times liters per minute = liters.
Furthermore, the area of the rectangle bounded by the graph of and the horizontal axis between and gives us the volume of water after minutes:
Function r sub 1 is graphed. A rectangular area under the line is shaded. The area extends from 0 to 6 minutes and from 0 to 5 liters per minute . The area of the rectangle is calculated as 6 minutes times 5 liters per minute = 30 liters.
Now say another tank is being filled, but this time the rate isn't constant:
Function r sub 2 is graphed. Time in minutes is on the x-axis, from 0 to 10. Rate, in liters per minute, is on the y-axis. The graph is a curve. The curve starts at (0, 0), moves upward concave down to about (5.2, 6), moves downward concave down, and ends at about (10, 0.8).
How can we tell the volume of water in this tank after minutes? To do that, let's think about the Riemann sum approximation of the area under this curve between and . For the sake of convenience, let's use an approximation where each rectangle is minute wide.
The previous function, r sub 2, is graphed. Six rectangular bars, each 1 unit, or 1 minute, wide rise vertically from the horizontal axis to the curve from 0 to 6 minutes. Each bar moves upward so that its top right vertex touches the curve. The top left vertex for the five rectangles from 0 to 5 are outside of the curve. Each rectangle has less outside of the curve than the previous. The sixth is completely within the curve. From left to right, the rectangles have the following approximate heights. 1.8, 3.4, 4.7, 5.6, 6, 5.9.
We saw how each rectangle represents a volume in liters. Specifically, each rectangle in this Riemann sum is an approximation of the volume of water that was added to the tank at each minute. When we add all the areas, i.e. when all the volumes are accumulated, we get an approximation for the total volume of water after minutes.
As we use more rectangles with smaller widths, we will get a better approximation. If we take this to a limit of accumulating infinite rectangles, we will get the definite integral . This means that the exact volume of water after minutes is equal to the area bounded by the graph of and the horizontal axis between and .
Function r sub 2 is graphed. The area between the curve and the t-axis, between t = 1 and t = 6, is shaded.
And so, integral calculus allows us to find the total volume after minutes:
Definite integral of the rate of change of a quantity gives the net change in that quantity.
In the example we saw, we had a function that describes a rate. In our case, it was the rate of volume over time. The definite integral of that function gave us the accumulation of volume—that quantity whose rate was given.
Another important feature here was the time interval of the definite integral. In our case, the time interval was the beginning and minutes after that . So the definite integral gave us the net change in the amount of water in the tank between and .
These are the two ways we commonly think about definite integrals: they describe an accumulation of a quantity, so the entire definite integral gives us the net change in that quantity.
Why "net change" in the quantity and not simply the quantity?
Using the above example, notice how we weren't told whether there was any amount of water in the tank prior to . If the tank was empty, then is really the amount of water in the tank after minutes. But if the tank already contained, say, liters of water, then the actual volume of water in the tank after minutes is:
This is approximately .
Remember: The definite integral always gives us the net change in a quantity, not the actual value of that quantity. To find the actual quantity, we need to add an initial condition to the definite integral.
Problem set 1 will walk you through the process of analyzing a context that involves accumulation:
At time , a population of bacteria grows at the rate of grams per day, where is measured in days.
Function r is graphed. Time in days is on the x-axis, from 0 to 10. Growth rate, in grams per day, is on the y-axis. The graph is a curve. The curve starts at (0, 1), moves upward concave up through (8, 5), and ends at about (10, 7.3). The area between the curve and the x-axis, between t = 0 and t = 8, is shaded.
What are the units of the quantity represented by the definite integral ?
Common mistake: Using inappropriate units
As with all applied word problems, units play an important role here. Remember that if is a rate function measured in , then its definite integral is measured in .
For example, in Problem set 1, was measured in , and so the definite integral of was measured in .
Eden walked at a rate of kilometers per hour (where is the time in hours).
What does mean?
Common mistake: Misinterpreting the interval of integration
For any rate function , the definite integral describes the accumulation of values between and .
A common mistake is to disregard one of the boundaries (usually the lower one), which results in a wrong interpretation.
For example, in Problem 2, it would be a mistake to interpret as the distance Eden walked in hours. The lower boundary is , so is the distance Eden walked between the hour and the hour. Furthermore, in cases like that where the time interval is exactly one unit, we usually say "during the hour."
Julia's revenue is thousand dollars per month (where is the month of the year). Julia had made thousand dollars in the first month of the year.
What does mean?
Common mistake: Ignoring initial conditions
For a rate function and an antiderivative , the definite integral gives the net change in between and . If we add an initial condition, we will get an actual value of .
For example, in Problem 3, represents the change in the amount of money Julia made between the and the months. But since we added , which is the amount Julia had at the month, the expression now represents the actual amount in the month.
Connection with applied rates of change
In differential calculus, we learned that the derivative of a function gives the instantaneous rate of change of for a given input. Now we're going the other way! For any rate function , its antiderivative gives the accumulated value of the quantity whose rate is described by .
The function gives the amount of ketchup (in kilograms) produced in a sauce factory by time (in hours) on a given day.
What does represent?
Want more practice? Try this exercise.
Want to join the conversation?
- In cases where the time interval is exactly one unit (e.g. [2,3]), why do we usually say "during the third hour" instead of "during the second hour"?
EDIT: Specifically, in the example where Eden walks "between the 2nd hour and the 3rd hour", wouldn't it make more sense to say "during the 2nd hour" because the first statement implies that Eden walked from the start of the 2nd hour to the start of the 3rd hour?(21 votes)
- Think about if you were waiting for a friend. Waiting for your friend for the first hour would be from time 0-1, waiting for your friend for the second hour would be 1-2, and waiting for your friend for the third hour would be between 2-3. You may want to find a new friend though if he makes you wait that long.
Time in centuries is similar. Between the years 0-99 AD is the first century, 100-199 is the second century, etc.(139 votes)
- For the the calculation of the integral of r2(t)=6sin(0.3t) under "Thinking about accumulation in a real world context," why is the 6 divided by 0.3?
Sorry, the answer given under Ryan's comment is not loading. Can someone please explain it? Thank you(17 votes)
- Due to the chain rule, when you differentiate you would multiply by 0.3. When you integrate you reverse this process and so you divide by 0.3. You can make a u-substitution to make this easier to understand if you need to.(17 votes)
- For the the calculation of the integral of r2(t)=6sin(0.3t) under "Thinking about accumulation in a real world context," why is the 6 divided by 0.3?(14 votes)
- what is the difference between definate integral calculus and indefinate integral calculus(4 votes)
- In the field of integral calculus we speak of definite integrals and indefinite integrals.
In short, an indefinite integral is a function (𝐹(𝑥) + 𝐶),
while a definite integral is a value ("area under the curve").(8 votes)
- What does DT or dx means in this equation?(3 votes)
- dt, dx and any d-something in calculus means "a small change in this thing" In fact it means so small it's basically 0(5 votes)
- at prob 3...is d really the correct answer? wouldnt it rather be the total ammount of money she made during month 1 to 5.....plus 3?? i mean u dont really know if that 3 was the ammount made between 0-1? right(2 votes)
- "Julia's revenue is r(t) thousand dollars per month (where t is the month of the year). Julia had made 3 thousand dollars in the first month of the year."(3 votes)
- where did the .3 come from in sin.3 example(2 votes)
- 6sin(.3t) you mean? That was just part of the chosen rate that was desired to be shown.(2 votes)
- Why is the final problem 0~4 k′(t)dt, when the answer is, "The amount of ketchup produced over the first 4 hours"? To match that answer, shouldn't the integral be, 0~4 k(t)dt (while 0~4 k′(t)dt would be the accumulated rates of change over the first four hours -- or nothing at all if that notation does not apply to integrals)?(2 votes)
- What software do you use to make your graphs?(2 votes)
- you can use desmos graphing calculator or geogebra(1 vote)
- in problem 3, "between the 1st and the 5th months" means 4 months, right? From month 0 to month 1 is the 1st month.
in "Common mistake: Ignoring initial conditions" section, it also uses the term antiderivative, which hasn't been taught in this course.(1 vote)
- Ahh not quite. When we say months 1 to 5, we mean the ending of month 1 to the end of month 5 (which does still account for 4 months though). The "3" in the question is the income from the end of month 0 the end of month 1 though. It's kinda confusing notation, so I hope I explained it well enough😅
The antiderivative is pretty much the inverse of a derivative. If you apply the derivative to f(x), you get f'(x). And if you apply the antiderivative to f'(x), you get f(x) again (Pretty much how inverses work). The future lessons should cover it in depth (especially when you reach the "Fundamental Theorem of Calculus"). Just remember that the integral and the antidervative are different things. Many use them synonymously(3 votes)