Fundamental Theorem of Calculus II

This video looks at the second fundamental theorem of calculus, where we take the definite integral of a function whose anti-derivative we can compute. This gives a short and sweat way to compute any definite integral provided we can come up with an anti-derivative.

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Now it's your turn:
1) Summarize the big idea of this video in your own words
2) Write down anything you are unsure about to think about later
3) What questions for the future do you have? Where are we going with this content?
4) Can you come up with your own sample test problem on this material? Solve it!

Learning mathematics is best done by actually DOING mathematics. A video like this can only ever be a starting point. I might show you the basic ideas, definitions, formulas, and examples, but to truly master math means that you have to spend time - a lot of time! - sitting down and trying problems yourself, asking questions, and thinking about mathematics. So before you go on to the next video, pause and go THINK.

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This video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati.

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[madghostek]

A good way to think about the subtraction F(b)-F(a) is that F(b), as a function accumulaing area from some point, contains amount of area up to point b, while F(a) has area up to a. Area up to point a overlaps with area up to b, after subtracting we get the part without overlap, aka our area between a and b. Notice that we don't really know what the value of F(a) or F(b) exactly mean, because F is a family of functions... but we know after subtraction it will be the area the integral was asking for

[daviddavid]

Best explanation I've seen so far, thanks alot

[harirajagopal]

Brilliant summarization at the end! Thank you

[naiko]

Ah, there's lot of things going on in the background here! :)

In 'F(b) - F(a)' of course each of the two anti-derivatives can be expanded into an integral from a generic number 'n' up to 'b' and the other up to 'a' respectively. (I used 'n' instead of 'a' as usual because we are already using 'a').
A cool thing that can be seen this way is that if the chosen 'n' was equal to 'b', we would find that the integral from 'a' to 'b' is the same as the integral from 'b' to 'b', MINUS the integral from 'b' to 'a'. But 'b' to 'b' is 0. In other words we can invert the 2 extremes of integration by inverting the sign before the integral. Which makes sense if you think about the meaning of the integral with inverted 'a' and 'b'.

Furthermore, the rule F(b) - F(a) doesn't spawn from the ether, it should first of all make intuitive sense as a subtraction of areas under the curve. Secondly, it can indeed be derived from our definition of antiderivative by playing with the constant C and writing down some equations.

Also, notice that of course a definite integral doesn't ever produce any constant of integration, and that makes sense because it's only answering the question "what's the area under the curve? Give me a number". (Besides, when we do F(b) - F(a), yes, those are antiderivatives that we write, but their constants C will cancel out).
Pay attention and notice that an integral from 'a' to 'x' instead returns a function, and that's because something inside the integral depends on 'x', and 'x' is a variable, so the integral can't return a finite quantity, but has to answer with a function of the area under the curve!
If you happen to have weird flashbacks about the chain rule being applied to the integral from 'a' to 'x^3' in the previous video, well, remember that the only reason why we applied it is because we had 'x^3' as one of the boundaries of the integration, not just 'x'.

And yes, again, neither integrals from 'a' to 'b', neither ones from 'a' to 'x' have a reason to return a constant of integration C. In fact they do not!
An integral only returns the are under the curve, at most a function of the area under the curve, but an integral is NOT the general anti-derivative of a function, it's a specific anti-derivative at most (the one with c=0).
The general anti-derivative of a function returns an integral + C, and while yes, an integral + 0 ( where c=0) IS AN anti-derivative, there's also infinitely many others with a different C.

In short, the constant C is only about the antiderivatives, not the integrals!
Why is it confusing then?
Notation and terminology!
The INDEFINITE integral...what is that? It's the anti-derivative, it's been renamed! It's not an integral, like the definite integral from 'a' to 'b' or from 'a' to 'x', it's really, really, another way to write F(x) +C, the general anti-derivative, where F(x) = integral from 'a' to 'x'.

Finally, computationally, when calculating the anti-derivative F(x) in a specific 'x', surely we can just plug in the corresponding anti-derivative/primitive function, substitute x and compute, but what the hell are we doing?
Isn't F(x) = integral from 'a' to 'x' ??? what does it even mean when 'a' is not defined? Doesn't it feel weird?
Well, the thing is, we previously proved that it works for any 'a' we might choose! This was when we first found out that the general anti-derivative is equal to the integral + C, if you go back to the derivation of the formula you really see that the 'a' of the anti-derivative really doesn't matter.
And it makes sense intuitively too. When we first defined the anti-derivative F ' (x) = f(x) you can clearly see that the only thing that matters is the derivative/amount-of-change of F(x), not the actual values it takes, so the 'a' we start from to count the area is totally irrelevant.

Thanks to the Fundamental Theorem of Calculus we can now calculate the definite integral way more easily, thanks to the connection to anti-derivatives, without going through Riemann Sums, and obtain the perfect area values.

[sonjatoutenhoofd]

Thank you! I watched several times and understood your explanation

[raykos]

In our class we learned this as the first fundamental theorem of calculus.

[journeymantraveller]

More an explanation that FTC 2 is saying "find an antiderivative to perform integration" than a proof. The proof of FTC 1 does give insight into the connection.

[kreskimatmy]

but why F(b) gives you area under curve, at the definition of Antiderivative there's nothing about areas!