Constants of Integration
In the previous activity we saw that
G(x), the antiderivative of g(x), is a model of car count on Route 15 in Johnson Vermont. Specifically, we saw that G(x) is a model of the car count on the highway between midnight and minute x. However I bet you feel a little confused about the negative 572.31312 at the end of G(x). Let's take a closer look at this number, which is called the Constant of Integration.
To understand what's going on, let's first take look at a slightly simpler function. Below is the visualization of the integral of f(x)=3x^2+2 which we saw earlier. As we saw, the antiderivative of f(x) is F(x)=x^3+2x. The only difference is that now I've added a slider for a number c in the right panel. Slide c and see what changes but also pay attention to what does not change.In the applet,
c is added to G(x). That's it. It doesn't show up anywhere else. Consequently, c translates G(x) up if c is positive, and down if c is negative. This is the full extent of the impact of the number c.
Now let's discuss what c does not impact.
- First, because of Monkey Rule 2, the derivative of
G(x)will always beg(x)no matter whatcis. - Second,
chas no impact ong(x)in the left panel, and therefore has no impact on the integral ofg(x), which is visualized as the red area. - Finally, and perhaps most interestingly, the addition of
chas no impact on the difference betweenG(b)andG(a), visualized by the red segment. Since bothG(b)andG(a)are slid up or down by the same amount, their difference remains the same, and is entirely independent ofc.
c added to G(x) is called the constant of integration. It's traditional to use the letter "c". Anytime you find an antiderivative, it's customary in algebra based calculus courses to add a constant of integration by appending "+c" to the end of the antiderivative.
For instance, in the applet above, the function to be integrated on the left is f(x)=3x^2+2, so it would be customary to write the antiderivative as F(x)=x^3+2x+c.
As we've seen, the addition of a constant of integration has no impact on the integrals of f(x). However, the addition of the constant of integration has a huge impact on the meaning of the antiderivative F(x) when it is a mathematical model. If you continue studying calculus, you will encounter something called a "differential equation" which is a very useful thing in constructing and studying mathematical models. In differential equations the constant of integration is of paramount importance.
For now though, let's see what the constant of integration means in the context of an antiderivative which is a mathematical model. Let's go back to the integral of the model of traffic rates on Route 15 in Johnson Vermont, g(x), from the previous activity. As we know, the antiderivative G(x) is a model for the accumulated effect of this traffic rate; in other words G(x) is a model of car count. The applet below is the same as the one from the previous activity, but instead of having 572.31312 subtracted from G(x), I've added created a constant c, set it to negative 572.31312, and added it to G(x). Furthermore, there is a slider for you to adjust c. Try it out.Just like before, it's clear that
c has no impact on either the integral of g(x) on the left, or the difference G(b)-G(a) on the right. The only impact of c is to translate the graph of the antiderivative G(x) up and down, but there is no impact on the red area (integral) on the left, or the red segment (difference) on the right.
So what impact does c actually have? Mathematically, all that c does is shift G(x) up and down, but as a mathematical model, c impacts the meaning of the antiderivative G(x). Specifically, it sets the "start time" of the antiderivative model.
Reset the app above so c goes back to negative 572.31312. Notice that the x-intercept of G(x) is x=0. The best way to think of the x-intercept of this model is as a start time. If you shift c even more negative, for example to about negative 800, you'll see that the x-intercept shifts closer to a. This is just an adjustment to the model in response to a questions such as: "what time do we want to start counting cars?" By shifting c to negative 800, we're saying, let's start counting a little later. We'd have to work out the algebra to see exactly what that time is, but you can see that's what is happening.
How would you work out that algebra? Well, once you pick a value for c, you would then solve for x by setting G(x) (with it's c added) equal to 0. Depending on how bad the algebra is, this could be easy or very very hard. It'd actually be very very hard for G(x) from this Route 15 model.
You might also wonder how I figured out to set c equal to negative 572.31312 so that the model would "start counting at midnight." That is actually a bit easier. Since I knew I wanted G(x) to be 0 when x is 0 (this is what it would mean for the model to "start counting at midnight"), I set c to 0, then set x to 0, and calculated G(0). Specifically, I calculated this value:
Bust out your TI-84, ask Siri, or just trust me: that is positive 572.31312. So to make G(x) equal to 0 when x is equal to 0, I just need to subtract 572.31312. And that's the entirety of how I selected c. That may seem hard at first, but do a few more mathematical modeling projects with integrals, and it will become more transparent.
Before we move on I have some good news and some bad news. Here's the bad news: there's not any very good way of summarizing the meaning of the constant of integration for every mathematical model. Every mathematical model that is the integral of something else (like G(x) above) will have a different meaning for c. The good news is though that the constant of integration of all mathematical models always has a lot of meaning, so you can usually figure it out by taking a few minutes and thinking it through! One thing is absolutely for sure though: despite what Instagram math memes say: you should absolutely not just ignore constants of integration!
In the next activity we'll take a much closer look at the constant of integration as well in a mega classic ★★★★★ application of integration: Falling Stuff on Earth.