Introduction

Welcome to the FiveHive article for Unit 6.4 of AP Calculus! 

In this article, we will be looking at two major components of calculus: The Fundamental Theorem of Calculus (FTC) and Accumulation Functions. 

As usual, we will only cover the topics included in the CED for unit 6.4.

The Fundamental Theorem of Calculus

The fundamental theorem of calculus is, well, fundamental. From here on out in this article, we will refer to this concept as the FTC for the sake of simplicity. 

So, what is it?

The FTC is a connection between derivatives (see units 2-5) and integration. If you want to sound fancy, the FTC connects differential calculus to integral calculus. This can be seen through these statements:

1st Fundamental Theorem: ${\int }_a^{b}f(x)dx=F(b)-F(a)$

2nd Fundamental Theorem: If $F(x)={\int }_a^{x}f(t)dt$, considering $f$ is continuous, then $\frac{d}{dx}\left[\frac{F(x)}{x}\right]=f(x)$.

NOTE: Some teachers may teach these in a flipped order (2nd theorem first and 1st theorem as second), but the idea behind them stays the same. 

For now, the understanding of these laws is enough for us to move on to accumulation functions.

Accumulation Functions

As we have seen with Riemann Sums, an integral is the area underneath a curve. This same idea is what we are applying when we define integrals as accumulation functions. 

An integral can be thought of as a Riemann Sum with an infinite amount of rectangles fitted underneath the curve. This can be seen in equation form as such: 

${\int }_a^{b}f(x)dx=\underset{n\to \infty }{\lim }\sum _{i=1}^{n}f(x_i^{\ast })\Delta x$

You do NOT need to know this if you are in AP Calculus AB, but it is a cool piece of information to know and it may also help with connecting topics to understand integration better. 

Now that we have a refresher of integrals, let us connect it to the FTC! 

Similar to how we described properties of a function $f$ using its derivative $f’$ in differential calculus, we will talk about functions and their antiderivatives in integral calculus. 

Here, we will need to use the second fundamental theorem of calculus. 

Example 1: 

Let us consider a function $g(x)={\int }_0^{x}f(t)dt$, where $f(t)=-t^2+4$. 

Using the 2nd FTC, we can say that $g$ is the antiderivative of $f$ and hence, $g’$ = $f$. 

Therefore, we can say that $\frac{d}{dx}g(x)=f(x)$.

In doing so, we find out that $f(x)=-x^2+4$.

Now, let us consider an interesting scenario. Instead of the upper limit being just $x$, what would we do if it was something like $2x$? The answer is to simply multiply the derivative of the upper limit to the final answer. 

Let’s look at an example:

Consider the same function $g(x)$, but this time it is equal to ${\int }_0^{2x}f(t)dt$. 

The first step is to take the derivative of the upper limit, which is $2$. Now that is done, we simply substitute $t$ with $2x$, as we have done with an upper limit of $x$. 

This yields:

$f(x)=2(-x^2+4)$

Practice

Okay, that was a lot. But hey, you’re in unit 6 of AP Calculus; let’s keep that momentum going with some practice problems!