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## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB>Unit 6

Lesson 4: The fundamental theorem of calculus and accumulation functions

# The fundamental theorem of calculus and accumulation functions

The fundamental theorem of calculus shows how, in some sense, integration is the opposite of differentiation. Created by Sal Khan.

## Want to join the conversation?

• can someone help me, i still don't understand this. i'm in yr 11 right now and i have watched this over and over and i still don't get it. if someone could explain and make it simpler. much appreciated . •   Many students find this confusing. Part of the problem is that in almost all our other work we're looking at a graph of the function we're studying. In this case, we're studying a function F(x) but looking at a graph of a different function, f(t). So the first thing I would offer in trying to understand this better is to get a clear picture that this graph does not depict F(x). Instead, F(x) is the area under this graph between point a and point x. As x moves to the right, this area increases, even if f(t) is decreasing. The graph doesn't show F(x) at all; in fact, it doesn't have an x-axis. We're given a function f(t) and asked to think about another function, not displayed, that is the area under the curve, not the value displayed on the curve.

If you get that much firmly in mind, the rest should be easier, but there are a couple of other points of confusion. We're trying to show that if we have a function F(x) that provides the area under the curve f(t), the derivative of F(x) is f(x). Some people have a hard time with this duality of f(t) and f(x). The key to understanding this is to realize that f, by itself, is a function. It can operate on t or x or any other variable or constant within its domain. We set it up initially as f(t) because we're using x for a different purpose (area under the curve) and it would create a logical contradiction for x to be both the independent variable that produces the curve and the independent variable that produces the area under the curve. We use t in place of x for the curve, and sometimes this is called a dummy variable.

The other thing that many people find confusing is that we get the same result no matter what point we choose for a, the starting point of the region we're studying. The reason is that we're trying to get at the rate of change in F(x), and the rate at which F(x) is changing doesn't depend on how much area there is to the left of x.

Ultimately we're trying to prove something that is, in a way, quite obvious: that the rate of change in the area under a curve is given by the value of the curve. When the curve is at y = 3, for example, the area under the curve is expanding at a rate of 3. Makes sense, right?
• When defining Fundamental Theorem of Calculus shouldn't the function of lower limit of the integral be zero? Shouldn't there be '-f(a)'' after f(x) in the right hand side otherwise? •   No, as long as a is not dependent on x.

Assuming that it isn't, you would have d/dx( integral |a to x| f(t)dt)=d/dx (F(x)-F(a))=f(x), because a is a constant and so F(a) is also a constant which, when you take the derivative, becomes 0.
• I was taught in class that this is the second fundamental theorem of calculus, and after searching online I got the same response. Am I missing something here? •   No, you're not missing a thing. :) The Fundamental Theorem of Calculus has two parts. Many mathematicians and textbooks split them into two different theorems, but don't always agree about which half is the First and which is the Second, and then there are all the folks who keep it all as one big theorem. In practice, you can figure out which part people mean by context, so it isn't as confusing as you might fear.
• why is it in terms of f(t), why not f(x)? •   We need to use a variable other than x for the function because we're using x for a point between a and b. Using x for both purposes at the same time would create a logical inconsistency. Bear in mind that we're speaking loosely when we say f(x) is a function. The function is just f, and it produces the same result whether we feed x or t or something else into it. In this context, t is sometimes called a dummy variable because we aren't really interested in t but need something to stand in for x to prevent the logic from breaking down.
• Why lower limit 'a' of integral doesn't show the result differentiating the integral? In other words, why 'a' doesn't influence the calculations? •  That's a good question. In fact, I suspect it gets asked in just about every calculus class. One way to answer is that we're dealing with a derivative of a function that gives the area under the curve. Changing the starting point ("a") would change the area by a constant, and the derivative of a constant is zero. Another way to answer is that in the proof of the fundamental theorem, which is provided in a later video, whatever value we use as the starting point gets cancelled out. We'll compare the value at point x with the value at point x + Δx. Both of those values will include a subtraction for the value at a, so when you subtract one from the other, these values cancel out.
• there are an infinite number of infinitely small time segments between any two moments in time. So I'm wandering how time still flows from a second to the next? I've been studying calculus and just wandering. •  These are mathematical models of reality, not reality itself.

While this is not a well-confirmed idea, and many scientists disagree, there are some scientists who think that there is a smallest unit of time, something like the time equivalent of an atom -- you just can't get any briefer a period of time. This unit is the Planck time, which is equal to about 5.39×10⁻⁴⁴ sec.

What is fairly well accepted by scientists is that it is impossible by any means to measure time shorter than 1 Planck time, whether or not there is such a thing as part of a Planck time. So, you could only ever measure whole number increments of a Planck time. But, the dispute amongst scientists is not over the measurement, but over whether there even exists a time period shorter than 1 Planck time.

At present, we don't have the technology to measure time anywhere near as brief as a Planck time (though we have other reasons for knowing that is the limit of measurement). And, if current scientific hypotheses are correct, we will never get close to measuring time that small.

So, while we can create a mathematical model and imagine cutting it into infinitely many pieces just so we can minimize our approximation errors, that doesn't mean that something in the real world can be cut up into infinitely many slices.
• at sal says that every continous function has an anti-derivative.but there are continous functons like sin(x)/x which is undefined only at x=0 .but still it is not possible to find the anti-derivative of this function over any interval which excludes 0....why is that? • The function sin(x)/x is famous in many ways. But it is not continuous for all x in the real numbers, as pointed out above. At x = 0, the function is undefined. So Sal's statement about continuous functions is correct.
Of course you can take a limit of a seemingly undefined function using the squeeze theorem, l'Hopital's rule, or by using complex variables or other tricks found in multivariable calculus and beyond.
A great summary of sin(x)/x can be found here:
press.princeton.edu/books/maor/chapter_10.pdf
• Um....I'm a bit confused. Isn't derivative the slope of the tangent line at a point in a curve? And isn't integral an area under a curve? So how can we have a derivative of an integral - a "slope of tangent line" of an "area"? Sorry....I'm just really new to calculus....and I just keep trying to visualize everything :-) • You can interpret the derivative as the "instantaneous rate of change". Thus the derivative of the integral is the "instantaneous rate of change of the area under the curve" as the upper bound on the integral advances.

In theory you could produce a graph to represent the area under the curve as a function of the upper bound, and talk about the slope of that graph, but I've rarely done so.  