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[music] There are some properties of the integral that are worth summarizing.

Â For example, here's a property of integration that's often useful.

Â If I want to integrate from a to b, right? That's what this last integral is.

Â And it's the area under the graph between a and b.

Â It's this whole region here that I've colored in.

Â But I could split that up into two separate integrals, right?

Â I could integrate from a to c at this first red integral, and then I could

Â integrate from c to b, that's the second blue interval.

Â And if I add together those two areas, I get the total area from a to b, right?

Â And that geometric fact is exactly what's written down here, with symbols using our

Â fancy integration notation. There's also a constant multiple rule.

Â Well, here's the constant multiple rule. For some constant K, the integral from a

Â to b of that constant times a function is that constant times the integral of that

Â function. And this also makes sense geometrically.

Â Here's two pictures. Here's the graph of y equals f of x.

Â Here's the graph of y equals K times f of x.

Â This thing here calculates this area, the area under the graph of K times f of x.

Â And it's K times just the area under the graph of f of x.

Â And it makes sense. Because if you take this graph and your

Â stretch it K times, that multiplies the area by a factor of K.

Â What about the intergral of a sum? Well, the integral of f of x plus g of x

Â from a to b, right? That's this total area here.

Â It's related to the area under the graph of f and the area under the graph of g,

Â right? It's related to these integrals.

Â Here, in green, I've sort of demonstrated what the area under the graph of f of x

Â looks like with a specific Riemann sum. And in here, in red, I've drawn some

Â rectangles for the Riemann sum of g. But they're, you know, sort of shifted up

Â a bit. Because this curve here is the graph of y

Â equals f of Xx plus g of x. So, the heights of these rectangles are

Â actually what I would get if I were to just integrate g of x, right?

Â The distance between f of x plus g of x, and f of x is exactly g of x here in red.

Â So, this is kind of a proof by stacking, if you like, that the integral of f plus g

Â is the integral of f plus the integral of g.

Â A lot of these rules have analogs for the sigma notation stuff.

Â We had this rule that said I could have pasted together integrals.

Â And there's a corresponding rule for sum that says, if I sum f of the numbers

Â between 1 and m, and then f of the numbers between m plus 1 and K, that's the same as

Â applying f to all the numbers between 1 and K and adding that up, right?

Â So, this same kinds of rules, I mean, there's an analogy there.

Â Same kind of game here, right? I've got this constant multiple rule for

Â integrals and I've got a corresponding constant multiple rule for sums.

Â Of course, this constant multiple rule is just called distributivity, right?

Â If I add up K times something, that's K times the sum of these things.

Â But, it's the same kind of rule, right? I had this formula that said the integral

Â of the sum is the sum of the integrals. We've got the same kind of formula for a

Â sum, right? If I take a sum of f of n plus g of n,

Â that's the same as adding up f of n for all the numbers between a and b.

Â And then, adding to that g of n for all the numbers between a and b.

Â And we're also seeing some similarities to the rules for derivatives.

Â I've got the constant multiple rule for integrals.

Â I've got a constant multiple rule for sums and I've got a constant multiple rule for

Â derivatives. The derivative of a constant times some

Â functions that constant times the derivative of the function.

Â 3:56

Same kind of deals for sums, right? I've got this sum of the integrals is the

Â integral of the sum. I've got this sum of a sum is the sum of

Â the sums. And I've got the derivative of a sum is

Â the sum of derivatives. Fundamentally, mathematics is not just

Â about these rules, right? It's about the relationships between all

Â of these rules. We're seeing various objects now,

Â integrals, derivatives, the sigma notation.

Â And they're all sharing some common rules, right?

Â And working out those relationships is really part of the fun.

Â