[music] There are some properties of the integral that are worth summarizing.
For example, here's a property of integration that's often useful.
If I want to integrate from a to b, right? That's what this last integral is.
And it's the area under the graph between a and b.
It's this whole region here that I've colored in.
But I could split that up into two separate integrals, right?
I could integrate from a to c at this first red integral, and then I could
integrate from c to b, that's the second blue interval.
And if I add together those two areas, I get the total area from a to b, right?
And that geometric fact is exactly what's written down here, with symbols using our
fancy integration notation. There's also a constant multiple rule.
Well, here's the constant multiple rule. For some constant K, the integral from a
to b of that constant times a function is that constant times the integral of that
function. And this also makes sense geometrically.
Here's two pictures. Here's the graph of y equals f of x.
Here's the graph of y equals K times f of x.
This thing here calculates this area, the area under the graph of K times f of x.
And it's K times just the area under the graph of f of x.
And it makes sense. Because if you take this graph and your
stretch it K times, that multiplies the area by a factor of K.
What about the intergral of a sum? Well, the integral of f of x plus g of x
from a to b, right? That's this total area here.
It's related to the area under the graph of f and the area under the graph of g,
right? It's related to these integrals.
Here, in green, I've sort of demonstrated what the area under the graph of f of x
looks like with a specific Riemann sum. And in here, in red, I've drawn some
rectangles for the Riemann sum of g. But they're, you know, sort of shifted up
a bit. Because this curve here is the graph of y
equals f of Xx plus g of x. So, the heights of these rectangles are
actually what I would get if I were to just integrate g of x, right?
The distance between f of x plus g of x, and f of x is exactly g of x here in red.
So, this is kind of a proof by stacking, if you like, that the integral of f plus g
is the integral of f plus the integral of g.
A lot of these rules have analogs for the sigma notation stuff.
We had this rule that said I could have pasted together integrals.
And there's a corresponding rule for sum that says, if I sum f of the numbers
between 1 and m, and then f of the numbers between m plus 1 and K, that's the same as
applying f to all the numbers between 1 and K and adding that up, right?
So, this same kinds of rules, I mean, there's an analogy there.
Same kind of game here, right? I've got this constant multiple rule for
integrals and I've got a corresponding constant multiple rule for sums.
Of course, this constant multiple rule is just called distributivity, right?
If I add up K times something, that's K times the sum of these things.
But, it's the same kind of rule, right? I had this formula that said the integral
of the sum is the sum of the integrals. We've got the same kind of formula for a
sum, right? If I take a sum of f of n plus g of n,
that's the same as adding up f of n for all the numbers between a and b.
And then, adding to that g of n for all the numbers between a and b.
And we're also seeing some similarities to the rules for derivatives.
I've got the constant multiple rule for integrals.
I've got a constant multiple rule for sums and I've got a constant multiple rule for
derivatives. The derivative of a constant times some
functions that constant times the derivative of the function.
Same kind of deals for sums, right? I've got this sum of the integrals is the
integral of the sum. I've got this sum of a sum is the sum of
the sums. And I've got the derivative of a sum is
the sum of derivatives. Fundamentally, mathematics is not just
about these rules, right? It's about the relationships between all
of these rules. We're seeing various objects now,
integrals, derivatives, the sigma notation.
And they're all sharing some common rules, right?
And working out those relationships is really part of the fun.