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18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 23
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PROFESSOR: Today we're going to hold off just a little bit on boiling water. And talk
about another application of integrals, and we'll get to the witches' cauldron in the
middle. The that I'd like to start with today is average value. This is something that I
mentioned a little bit earlier, and there was a misprint on the board, so I want to
make sure that we have the definitions straight. And also the reasoning straight. This
is one of the most important applications of integrals, one of the most important
examples.
If you take the average of a bunch of numbers, that looks like this. And we can view
this as sampling a function. As we would with Riemann's sum. And what I said last
week was that this tends to this expression here, which is called the continuous
average. So this guy is the continuous average. Or just the average of f. And I want
to explain that, just to make sure that we're all on the same page. In general, if you
have a function and you want to interpret the integral, our first interpretation was
that it's something like the area under the curve. But average value is another
reasonable interpretation. Namely, if you take equally spaced points here, starting
with x0, x1, x2, all the way up to xn, which is the left point b, and then we have
values y1, which = f ( x1), y2, which = f ( x2), all the way up to yn, which = f ( xn).
And again, the spacing here that we're talking about is b - a / n. So remember that
spacing, that's going to be the connection that we'll draw.
Then the Riemann sum is y1 through yn, the sum of (y1 ... yn) delta x. And that's
what tends, as delta x goes to 0, to the integral. . The only change in point of view if
I want to write this limiting property, which is right above here, the only change
between here and here is that I want to divide by the length of the interval. b - a. So
I will divide by b - a here. And divide by b - a over here. And then I'll just check
what this thing actually is. Delta x / b - a, what is that factor? Well, if we look over
here to what delta x is, if you divide by b - a, it's 1 / n. So the factor delta x / b - a =
1 / n. That's what I put over here, the sum of y1 through yn / n. And as this tends to
0, it's the same as n going to infinity. Those are the same things. The average value