Integration and Accumulation of Change

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Onto unit 6 of AP Calculus! If you were asked to calculate how many miles a car traveled in 3 hours moving at a constant speed of 65 miles per hour, how would you do it? Using this example, you’ll learn about Riemann sums, the basics of integral calculus, and how to apply it to real life!

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#🏎️ Change Over Time

Accumulation of change is exactly what it sounds like—the sum, over time, of how much something has changed. Let’s return to the example at the beginning: we want to know how far a car has traveled, and we are given its speed (in miles per hour) and the length of time it has been traveling. We want to know how many miles it has accumulated over 3 hours. This problem is fairly straightforward, and we can solve it with a simple equation:

$$d=v\cdot t$$

where d is distance, v is velocity (speed), and t is time. Let’s apply it to the problem.

$$d=\frac{65\text{ miles}}{\text{hour}}\cdot3\text{ hours}=\boxed{195 \text{ miles}}$$

Notice that because we have the unit hours in the denominator of one term and the numerator of the other, they cancel, leaving us with just a unit of miles. This is how units will work for all of these problems—they should be the unit of the rate of change multiplied by the independent variable.

We’ve solved our first problem! Now, let’s connect it to calculus.

#📈 Graphing Change Over Time

We can depict the relationship between how fast, how long, and how far the car traveled with a graph.

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Graph representing the distance traveled by the example car.

Image courtesy of Emery

Here, we can see that the x-axis represents how long the car has been traveling and the y-axis represents how fast it is traveling. The shaded area under the curve represents the total distance traveled.

This example is about a car that is traveling at a constant speed, as though it was on the highway using cruise control. But in reality, a whole trip for a vehicle involves variation in speed. Let’s look at another example, where a car leaves its house, accelerates to 50 miles per hours, drives on the highway for almost two hours, decelerates to 20 miles per hour, finishes driving to its destination, and then decelerates to 0 miles per hour. How far did the car travel?

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Graph depicting the second example problem.

Image courtesy of Emery

Based on our simpler example, we know that the shaded area under the curve will give us the distance the car traveled. But, we can’t just use the equation for an area of a rectangle here, because it’s a much more complex shape! The following piecewise function describes the curve:

f(x) = \begin{cases}
5000x^2 & \text{if }0 \leq x \leq 0.1 ...