Professor Dave here, I wanna tell you about the fundamental theorem of calculus.
We just learned what integration is.
We can consider it the act of finding the area under a curve.
But what does this have to do with differentiation?
To see how these two operations are intertwined, we have to learn how to integrate functions
mathematically, using a specific algorithm.
To do this, we must introduce some new notation, so let’s learn these symbols now.
In the previous tutorial we saw that as we add up all our tiny rectangles, we get this
expression, with the sum of f of xi times delta x from i equals one to n, in the limit
of n approaching infinity.
Without getting stuck on all these symbols, let’s just think of this as an expression
that means “add up all the rectangles”.
The new way that we will represent this idea looks like this.
This weird symbol, introduced by Leibniz, is called an integral sign, and it looks kind
of like a long S because it represents a limit of sums.
The function being integrated, f of x, is the integrand, and these two numbers by the
integral sign are called the limits of integration.
A is the lower limit and b is the upper limit.
These tell us where to start and stop adding up rectangles.
The symbol dx, which represents the infinitesimal, is required for integration, but it has no
meaning by itself.
The function being integrated simply must be followed by dx in order for integration
to be possible.
Because this integral indicates a specific interval, it’s called a definite integral,
and it can be thought of as a number, a number much like the area under a curve.
Let’s learn how to compute integrals now.
The new algorithm we must learn is as follows.
If we want to integrate this function f of x over the interval from a to b, that will
be equal to F of b minus F of a, where the upper case F represents the antiderivative
of the function.
So what’s an antiderivative?
Well that’s a function for which if you take the derivative, you get the original
function, kind of like how squaring a square root gives you what’s inside the root, so
capital F is the antiderivative of lower case f, because big F prime equals little f.
In order to get an antiderivative, we are going to have to employ precisely the opposite
algorithm of the one that is employed during differentiation.
Remember that when we take a derivative, we bring this exponent down here, and then reduce
the exponent by one.
That means that to get an antiderivative, we have to increase the exponent by one, and
then divide the resulting term by the new exponent.
For example, to get the antiderivative of x squared, we must write x cubed over three.
We can verify that this is correct by taking the derivative.
Three comes down here, and the exponent goes to two.
The threes cancel, and we are left with x squared, which is what we started with.
So to contextualize this new definition of integration, say we bring back our x squared
curve and integrate from zero to one.
That will be equal to the antiderivative evaluated at one minus the antiderivative evaluated
at zero.
That’s what is meant by this notation, with either a vertical line or a closed bracket,
it means evaluate this function for the number on top, and then subtract from that the evaluation
for the number on the bottom.
When x equals one, x cubed over three equals one third, and when x equals zero, it equals
zero, so the answer is one third.
As we recall from the previous tutorial, this was the area under the curve that we got from
the tiny rectangles, as the number of rectangles increased to infinity.
So we can see that this approach of evaluating antiderivatives is doing exactly the same thing.
We are getting the area under the curve.
This was the incredible light bulb moment experienced by Newton and Leibniz, and their
contemporaries in the 17th century.
This was the realization that just like addition and subtraction, or multiplication and division,
differentiation and integration are inverse operations.
If we draw a ladder, and on each rung we write a multiple of 2, going up we add two every
time, and going down we subtract two every time.
Addition and subtraction are therefore inverse operations.
If we instead write the powers of two, then up represents multiplying by two, and down
represents dividing by two.
Multiplication and division are therefore inverse operations.
By precisely the same logic, if we write a series of functions on these rungs, each function
is the derivative of the one above, and the integral of the one below.
The downward direction represents differentiation and the upward direction represents integration.
If we take the derivative of this, we get the one below.
If we integrate this, we can get the one above.
And this simple fact, the inverse relationship of these two operations, is the essential
truth that we refer to as the fundamental theorem of calculus.
This new understanding means that integration can now be thought of not as the obsolete
approach of constructing tiny rectangles and finding the limit of a sum, but rather as
an entirely new computational algorithm that is simply the inverse of differentiation.
The derivative of the integral of a function is just that function, in the same way that
the square of a square root gives us what’s inside the root.
Integral calculus and differential calculus are united at last.
Now that we have a basic grasp of this theorem, let’s express it a little more formally.
We have some function f of t which is continuous over the interval from a to b.
If we want to integrate this function from a to some interval x, we get a new function,
g of x, equal to the integral of f of t dt from a to x.
Notice that this new function does not depend on t, but only x, the upper limit of integration.
If x is a specific number, instead of getting a new function, we would use the antiderivative
of f and evaluate for the limits of integration to get a number.
This would be the area of this section under the curve.
If we leave x as a variable, then we instead just get the function g of x.
We can think of g of x as the area under some section of the curve, depending on x, or the
“area so far” as x moves along the curve.
As we just learned, it will be the case that g prime of x will equal f of x, and that means
that g of x must be the antiderivative of f of x.
This brings us to the second part of the fundamental theorem, which is the algorithm we mentioned earlier.
The integral of some function, f of x dx from a to b, equals the antiderivative of f, upper
case F, evaluated over this interval, which can also be expressed as the antiderivative
of b minus the antiderivative of a.
This is our new way of approaching the evaluation of integrals, so we won’t be talking about
rectangles anymore.
Now that we have this theorem under our belts, we are ready to practice evaluating definite
integrals, so let’s move on to that concept next.