In the [url=https://www.geogebra.org/m/x39ys4d7#material/tx9hrc7k]previous activity[/url] we saw that [code]G(x)[/code], the antiderivative of [code]g(x)[/code], is a model of car count on Route 15 in Johnson Vermont. Specifically, we saw that [code]G(x)[/code] is a model of the car count on the highway between midnight and minute [code]x[/code]. However I bet you feel a little confused about the negative [code]572.31312[/code] at the end of [code]G(x)[/code]. Let's take a closer look at this number, which is called the [b]Constant of Integration[/b].[br][br]To understand what's going on, let's first take look at a slightly simpler function. Below is the visualization of the integral of [code]f(x)=3x^2+2[/code] which we saw earlier. As we saw, the antiderivative of [code]f(x)[/code] is [code]F(x)=x^3+2x[/code]. The only difference is that now I've added a slider for a number [code]c[/code] in the right panel. Slide [code]c[/code] and see what changes but also pay attention to what [i]does not change[/i].

In the applet, [code]c[/code] is added to [code]G(x)[/code]. That's it. It doesn't show up anywhere else. Consequently, [code]c[/code] translates [code]G(x)[/code] up if [code]c[/code] is positive, and down if [code]c[/code] is negative. This is the full extent of the impact of the number [code]c[/code]. [br][br]Now let's discuss what [code]c[/code] does [i]not[/i] impact. [br][br][list][*]First, because of [url=https://www.geogebra.org/m/x39ys4d7#material/rhupsdff]Monkey Rule 2[/url], the derivative of [code]G(x)[/code] will always be [code]g(x)[/code] no matter what [code]c[/code] is. [/*][*]Second, [code]c[/code] has no impact on [code]g(x)[/code] in the left panel, and therefore has no impact on the integral of [code]g(x)[/code], which is visualized as the red area. [/*][*]Finally, and perhaps most interestingly, the addition of [code]c[/code] has[i] no impact[/i] on the difference between [code]G(b)[/code] and [code]G(a)[/code], visualized by the red segment. Since both [code]G(b)[/code] and [code]G(a)[/code] are slid up or down by the same amount, their difference remains the same, and is entirely independent of [code]c[/code].[br][/*][/list]This number [code]c[/code] added to [code]G(x)[/code] is called the [b]constant of integration[/b]. It's traditional to use the letter "[code]c[/code]". Anytime you find an antiderivative, it's customary in algebra based calculus courses to add a constant of integration by appending "[code]+c[/code]" to the end of the antiderivative. [br][br]For instance, in the applet above, the function to be integrated on the left is [code]f(x)=3x^2+2[/code], so it would be customary to write the antiderivative as [code]F(x)=x^3+2x+c[/code]. [br][br]As we've seen, the addition of a constant of integration has no impact on the integrals of [code]f(x)[/code]. However, the addition of the constant of integration has a huge impact on the [i]meaning[/i] of the antiderivative [code]F(x)[/code] 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.[br][br]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,[code] g(x),[/code] from the [url=https://www.geogebra.org/m/x39ys4d7#material/tx9hrc7k]previous activity[/url]. As we know, the antiderivative [code]G(x)[/code] is a model for the accumulated effect of this traffic rate; in other words [code]G(x)[/code] is a model of car count. The applet below is the same as the one from the previous activity, but instead of having [code]572.31312[/code] subtracted from [code]G(x)[/code], I've added created a constant [code]c[/code], set it to negative [code]572.31312[/code], and added it to [code]G(x)[/code]. Furthermore, there is a slider for you to adjust [code]c[/code]. Try it out.

Just like before, it's clear that [code]c[/code] has no impact on either the integral of [code]g(x)[/code] on the left, or the difference [code]G(b)-G(a)[/code] on the right. The only impact of [code]c[/code] is to translate the graph of the antiderivative [code]G(x)[/code] up and down, but there is no impact on the red area (integral) on the left, or the red segment (difference) on the right.[br][br]So what impact does c actually have? Mathematically, all that c does is shift [code]G(x)[/code] up and down, but as a mathematical model, [code]c[/code] [i]impacts the meaning of the antiderivative[/i] [code]G(x)[/code]. Specifically, it sets the "start time" of the antiderivative model. [br][br]Reset the app above so [code]c[/code] goes back to negative [code]572.31312[/code]. Notice that the x-intercept of [code]G(x)[/code] 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 [code]c[/code] 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.[br][br]How would you work out that algebra? Well, once you pick a value for c, you would then solve for x by setting [code]G(x)[/code] (with it's [code]c[/code] 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 [code]G(x)[/code] from this Route 15 model.[br][br]You might also wonder how I figured out to set c equal to negative [code]572.31312[/code] so that the model would "start counting at midnight." That is actually a bit easier. Since I knew I wanted [code]G(x)[/code] to be 0 when x is 0 (this is what it would mean for the model to "start counting at midnight"), I set [code]c[/code] to 0, then set x to 0, and calculated [code]G(0)[/code]. Specifically, I calculated this value:[br][br][math]G\left(0\right)=5.96076\cdot0-\frac{5.10899}{0.00429}\cos\left(0.00429\cdot0-2.0721\right)+0[/math][br][br]Bust out your TI-84, ask Siri, or just trust me: that is positive [code]572.31312[/code]. So to make [code]G(x)[/code] equal to 0 when x is equal to 0, I just need to subtract [code]572.31312[/code]. And that's the entirety of how I selected [code]c[/code]. That may seem hard at first, but do a few more mathematical modeling projects with integrals, and it will become more transparent.[br][br]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 [code]G(x)[/code] above) will have a different meaning for [code]c[/code]. 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 [url=https://www.instagram.com/p/BXSYN3AHSYF/]Instagram math memes[/url] say: you should absolutely not just ignore constants of integration![br][br]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.