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

Lesson 7: The fundamental theorem of calculus and definite integrals

# Proof of fundamental theorem of calculus

The fundamental theorem of calculus is very important in calculus (you might even say it's fundamental!). It connects derivatives and integrals in two, equivalent, ways:
$\begin{array}{rl}I.& \phantom{\rule{0.167em}{0ex}}\frac{d}{dx}{\int }_{a}^{x}f\left(t\right)\phantom{\rule{0.167em}{0ex}}dt=f\left(x\right)\\ \\ II.& \phantom{\rule{0.167em}{0ex}}{\int }_{a}^{b}\phantom{\rule{-0.167em}{0ex}}\phantom{\rule{-0.167em}{0ex}}f\left(x\right)dx=F\left(b\right)\phantom{\rule{-0.167em}{0ex}}-\phantom{\rule{-0.167em}{0ex}}\phantom{\rule{-0.167em}{0ex}}F\left(a\right)\end{array}$
The first part says that if you define a function as the definite integral of another function $f$, then the new function is an antiderivative of $f$.
The second part says that in order to find the definite integral of $f$ between $a$ and $b$, find an antiderivative of $f$, call it $F$, and calculate $F\left(b\right)-F\left(a\right)$.
The AP Calculus course doesn't require knowing the proof of this fact, but we believe that as long as a proof is accessible, there's always something to learn from it. In general, it's always good to require some kind of proof or justification for the theorems you learn.

## First, we prove the first part of the theorem.

Proof of fundamental theorem of calculusSee video transcript

## Next, we offer some intuition into the correctness of the second part.

Intuition for second part of fundamental theorem of calculusSee video transcript

## Finally, we prove the second part of the theorem based on the first part.

The fundamental theorem of calculus and definite integralsSee video transcript

## Want to join the conversation?

• the first proof... thought a little about it and came up w/ an possible other way to solve it:

take it from the Riemann definition (right sum approx) by
b=x (upper bound)
d/dx(int[a;b]f(x)dx = d/dx(f(a+dx)dx+f(a+2dx)dx...n times...+f(b)dx)
as f(a+dx)dx+f(a+2dx)dx... etc are not x-dependent, the derivatives of all go to 0, so
d/dx(int[a;b]f(x)dx=d/dx(f(b)dx)=f(b)
seems reasonable for me, as it all relies on x being both, the upper bound (b) and the variable, as if i just put x=b, considering b as "the max{x}" in the defined interval [a;b]; besides it avoids all the f(t)dt mess that seems confusing.

so, i want to discuss this, even (probably), why is it wrong (didn't saw it anywhere), but seems quite intuitive for me, for dF/dx is just the last d(right area)=f(b)dx/dx + the local linearity stuff.
• Looks like you did a good job there, but I still can't understand what you did because it's hard to understand it when it's written in that way. Can you, or anyone else, please write it down on a piece of paper and upload a photo of it on the internet so everybody can comprehend what's going on?
• In the mean value theorem for integrals proof Sal uses the fundamental theorem of calculus and here in the first part he uses the mean value theorem. Isn't that a circular argument because it says that MVT is true from FTC and FTC is true from MVT?
• Yes.
But MVT can be proved independently from FTC. In fact, the most rigorous proofs are a bunch of 𝜖-𝛿 style computations, not a corollary from some other strong theorem.
• I understood the first part of the theorem this way. Integrating from a to x(between a and b) is like moving x-pivot from a to b. Along the way x -pivot moves, we add the y- value that we get from the function f(t) (Since F(t) is an integration.) So the F(t)'s derivative of the moment we enter the arbitrary x into f(t) must be +f(x), because F(x) is adding f(x) at that moment, and the derivative describes the rate of change at that moment. Is my intuition correct?
• If i understood well, you might be thinking that the function itself pivotes like in this simulation: https://bit.ly/2JK8toj (Desmos)

Tell my if it wasn't usefull :D
• I'm confused why we needed to prove that d/dx of F is f, when F was defined during creation as the area under f. When we (or Leibnitz I suppose I should say) specified the "width" as dx, doesn't that in itself say that the derivative of F, (the rate at which the area, F, is changing at x) is given by f(x)? Stated another way, by specifying the width as dx, which is negligible in the area calculation by design, doesn't our area change by f(x)? Perhaps this is not proof because it's recursive?
• That's true, but it isn't a rigorous proof. It's good that you have the intuition for it, though!
• at in the 1st video, the m.v.t. of def. integrals hasn't been taught so far.
• You're right. I think it's taught in Unit 8. I'd recommend getting back to the proof once you learn the theorem. But, if you want a shortened version of it to better grasp the proof here, it basically says that for a function f(x) defined on the interval [a,b], there exists some c such that the value of the function at that point equals the average value of the function. This should make intuitive sense because if the function takes on a range of values, at one point, it must take on the average value too. Observe the striking similarity with the MVT for derivatives, where the derivative at a point was equal to the average rate of change of the function.
• Would there be any critical errors in this analogy? (I came up with this to understand those theorems more intuitively.)

f = F'
f = productivity (speed in producing value)
F = wealth (accumulated value)
C = Inherited wealth
If productivity is static, wealth increases linearly
if productivity increases, wealth increases exponentially
-> INVEST IN EDUCATION!
• The basic idea is correct. Wealth would not necessarily increase exponentially, but it would certainly increase at a much faster rate.
• What about indefinite integrals? Those don't correspond to any area of a curve, yet they equal the anti-derivative of the function. How does that work?