R Lesson 29: An Integral Part of R Calculus

Hello everybody,

Michael here, and in today’s post, I’ll be discussing an integral part of R calculus-integrals (see what I did there?).

The integral facts about integrals

What are integrals, exactly? Well, we did spend the last two posts discussing derivatives, which are metrics used to measure the rate at which one quantity changes with respect to another quantity (i.e. like the change in Rotten Tomatoes critic scores from one MCU movie to the next as we discussed in this post-R Lesson 27: Introductory R Calculus). Integrals, on the other hand, measure rates of accumulation of a certain quantity over time.

Simple enough, right? Well, is it possible to think of integrals as reverse derivatives? Yes it is! I did mention that derivatives measure change of a given quantity from point A to point B while integrals measure the accumulation of a quantity over a given time period (which could be from point A all the way to point XFD1048576). In the context of mathematical functions, derivatives break up a function into smaller pieces to measure the rate of change at each point while integrals put the pieces of the function back together to measure the rate of change across the entire duration of the function (which obviously can be infinity)

Calculating integrals, the manual way, part 1

Before we dive into R calculations of integrals, let’s first see how to calculate integrals, the manual way, by hand.

Let’s take this polynomial as an example:

Now, how would we calculate the integral of this polynomial. Take a look at the illustration below:

Just so you know, the polynomial in green is the integral of the original polynomial. With that said, how did we get the polynomial 3/4x^4-2/3x^3+2x^2-7x+C as the integral of the original polynomial?

First of all, just as we did with derivatives, we would need to calculate the integral of each term in the polynomial one-by-one. To do so, you’d need to add one to the exponent of each term, then divide that term by the new exponent. Still confused? Let me explain it another way:

• The integral for 3x^3 would be 3/4x^4 since 3+1=4 and 3 divided by 4 is, well, 3/4.
• The integral for -2x^2 would be -2/3x^3 since 2+1=3 and 2 divided by 3 is, well, 2/3.
• The integral for 4x would be 2x^2 since 1+1=2 and 4 divided by 2 is 2.
• The integral for -7 would be -7x since constants have a power of 0 and 0+1=1 (and anything divided by 1 equals itself)
• You will likely have noticed an additional value in the integral that you may not be aware of-the constant C. I’ll explain more about that right now.

So, you may be wondering what’s up with the C at the end of the integral equation. Remember how earlier in this post I mentioned that you can think of derivatives as reverse integrals. You may recall that in our previous lesson-R Lesson 28: Another Way To Work With Derivatives in R-I discussed that during the process of calculating a derivative for a polynomial, the derivative of any constant in that polynomial is 0. This means that when finding the derivative of any polynomial, the constant disappears.

Now, since integrals can be considered as reverse derivatives, we should remember that when we integrate a polynomial, there was likely a constant that disappeared during differentiation (which I forgot to mention is the name of the process used to find the derivative of a polynomial). The C at the end of an integral represents the infinite number of possible constants that could be used for a given integral.

An integral illustration to this lesson

For my more visual learners, here is an illustration of how integrals work:

Just as a derivative would measure the change from one point to another in this curve, the integral would measure the area under the curve. The area in yellow represents the negative integral while the area in red represents the positive integral.

Still confused? Don’t worry-we’ll definitely go more in depth in this lesson!

Calculating integrals, the R way, part 2

OK, now that we’ve discussed how to calculate integrals the manual way, let’s explore how to calculate integrals the R way. You’ll notice that R won’t just spit out the integral of a given polynomial but rather calculate the integral using an upper and lower limit. Don’t worry-I’ll explain this more later, but for now, let’s see how the magic is done:

integral <- function(x) { 3*x^3-2*x^2+4*x-7 }
result <- integrate(integral, lower=1, upper=2)
result
5.583333 with absolute error < 6.6e-14

In this example, I’m showing you how R does definite integration. What is definite integration? Let me explain it like this.

In the previous section of this post, all we were trying to do was to calculate the integral polynomial of a certain expression. This is known as indefinite integration since we were simply trying to find the integration function of a given polynomial with the arbitrary constant C. As I mentioned in the previous section, the constant C could represent nearly anything, which means there are infinite possible integrals for any given polynomial.

However, with definite integration (like I did above), you’ll be calculating the integral at an upper and lower limit-this is certainly helpful if you’re looking for the integral over a specific range in the polynomial function rather than just a general integral, which can stretch for infinity. In R, to calculate the integral of a function over a given range, specify values for the lower and upper parameters (in this case I used 1 and 2). As you can see from the result I obtained, I got ~5.58 with an absolute error of 6.6e-14, which indicates a very, very, very small margin of error for the integral calculation. In other words, R does a great job with definite integration.

• Keep in mind that the integration calculation approach I discussed above will only work with a finite range of integration (e.g. lower=1, upper=2). It won’t work with an infinite range of integration (e.g. from negative infinity to positive infinity).

Plotting an integration function

Now that we know how to calculate integrals, the next thing we’ll explore is plotting the integration function. Here’s how we’d do so-using the polynomial from the first section and an integration range of (0,5):

integral <- function(x) { 3*x^3-2*x^2+4*x-7 }
integrated <- function(x) { integrate(integral, lower=0, upper=50)$value }
vectorIntegral <- Vectorize(integrated)
x <- seq(0, 50, 1)
plot(x,vectorIntegral(x), xlim=c(0,50), xlab="X-values", ylab="Y-values", main="Definite Integration Example", col="blue", pch=16)

So, how did I manage to create this plot? Let me give you a step-by-step explanation:

• I first set the function I wish to integrate as the value of the integral value.
• I then retrieved the integral of this function at the range (0,5). I also grabbed the value of the integral at this range and nested this result into its own function, which I then stored as the value of the integrated variable.
• I then vectorized the value of the integral at the range (0,5) and stored that value into the vectorIntegral variable.
• I then created an x-axis sequence to use in my plot that contained the parameters (0, 50, 1) which represent the lower limit, upper limit, and x-axis increment for my plot, respectively. This sequence is stored in the x variable.
• Last but not least, I used the plot() function to plot the integral of the polynomial 3x^3-2x^2+4x-7. One thing you may be wondering about is the x, vectorIntegral(x) parameter in the function. The x parameter gathers all the x values for the plot (in this case the integers 0 to 5) while the vectorIntegral(x) parameter calculates all of the correpsonding y-values for each possible x-value and gathers them into a vector, or array, for the plot.
  • Why choose vectorization to calculate the corresponding y-values? Well, it’s easier than looping through each possible x-value in the integral range to get the correpsonding y-values, since vectorization simply takes in all possible x-values (0-50 in this case) as the input array and returns an output array containing all possible y-values for each possible x-value (which in this case all seem to be between 4,000,000 and 5,000,000).

Calculating integrals, the manual way, part 3

So, now that I’ve shown you how to do definite integration the R way, let me show you how to do so the manual way. Let’s examine this illustation:

So in this illustration, I’m trying to calculate the integral for the polynomial 3x^3-2x^2+4x-7 using the (7,10) range. How do I do so. Well, first I perform some indefinite integration by finding the integral of the given polynomial-only thing here is that I don’t need the constant C. Next, since my integration range is (7,10), I evaluate the integral function for x=10 and subtract that result from the result I get after evaluating the integral function for x=7. After all my calculations are complete, I get 5342.25 as the value of my integral (rounded to two decimal places) at the integration range of (7,10).

• If you’re wondering what that weird-looking S means, that’s just a standard integral writing notation.
• To calculate the integral of any given expression for a given range, always remember to first find the integral of the polynomial and then evaluate that integral for x=both the upper and lower limits. Subtract the result of the upper limit evaluation from the result of the lower limit evaluation. And remember that, as we saw in our R integral calculations, there will always be a very very very small margin of error.
• In calculus function notation, the capital F represents the f(x) of the integral while the lowercase f represents the f(x) of that integral’s derivative.

Thanks for reading!

Michael