Averages

D. DeTurck

University of Pennsylvania

September 15, 2015

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Totals and averages

We have used integrals to calculate totals — the total areabetween two curves, the total volume of a solid, etc.One more kind of total:

Total mass: First, consider a long thin tube (so thin that it isessentially one-dimensional) lying along the x-axis between x = aand x = b. And suppose that the linear density of the material ofthe tube is δ(x) grams per centimeter. Then the mass of the littlebit of tube between x and x + dx is

dm = δ(x)dx

and so the total mass of the tube is

m =

ˆ b

aδ(x)dx .

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More mass

Next, consider the region in the plane between the two curvesy = f (x) and y = g(x) for a ≤ x ≤ b and suppose there is alamina (i.e., a thin, flat solid) with the shape of the region. Alsosuppose that the areal density of the material in the lamina varieswith x – say the density of the part of the region over x is δ(x)(grams per square centimeter or something like that). Then themass of the slice of region between x and x + dx is

dm = (f (x)− g(x))δ(x) dx

and so the total mass of the lamina is

m =

ˆ b

a(f (x)− g(x))δ(x) dx .

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3D mass

We can do the same with a solid of revolution around the x-axis ifthe density depends only on x . So if δ(x) is the density (in gramsper cubic centimeter this time), then the mass of the disk orwasher between x and x + dx is

dm = πf (x)2δ(x)dx or dm = π(f (x)2 − g(x)2)δ(x)dx .

So the total mass of the solid is

m =

ˆ b

aπf (x)2δ(x)dx or m =

ˆ b

aπ(f (x)2 − g(x)2)δ(x)dx .

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Averages and weighted averages

You probably learned the formula for the average value of thefunction f (x) for a ≤ x ≤ b:

avg =1

b − a

ˆ b

af (x)dx

Thinking about how this formula works will give us a way togeneralize it quite a bit.

To start, we’ll think of an example:

Section 201 202 203 204 205 206

Students 10 20 15 20 15 20

Avg. grade 8 7 8 9 8 6

The average grade is the average for each section. How can wecalculate the overall class average?

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Class average

Section 201 202 203 204 205 206

Students 10 20 15 20 15 20

Avg. grade 8 7 8 9 8 6

The challenge is that different sections have different numbers ofstudents. But we can get the average because, even though wedon’t know the individual students’ grades, we know for instancethat a total of 10 · 8 = 80 points were achieved by all the studentsin section 201, and 20 · 7 = 140 in section 202, etc. Since there are100 students altogether, we see that the overall class average is:

10 · 8 + 20 · 7 + 15 · 8 + 20 · 9 + 15 · 8 + 20 · 6100

=760

100= 7.6

The essential thing is that we knew the average in each part(section) of the class.

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More generally

We can calculate a weighted average of a set of quantities if wecan divide them up into groups (slices, disks, washers, shells,[infinitesimal] age brackets, etc.) for which

1 We know the average value of the quantity for each group

2 We know the size (area, volume, mass, population,etc) ofeach group

An example: Find the average value of x in the region in the firstquadrant bounded by y = 1− x2

Concentrate on the vertical slice, of width dx overx . Its area is (1− x2)dx and the average value ofx in the slice is simply x . So the total number of“x points” (like “exam points”) in the slice isx(1− x2)dx .

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Getting the average

An example: Find the average value of x in the region in the firstquadrant bounded by y = 1− x2

To get the average, we need to add up all the “x points” in all theslices for 0 ≤ x ≤ 1, and divide by the total area:

avg x =

ˆ 1

0x(1− x2)dx

ˆ 1

0(1− x2)dx

=1423

=3

8

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The average y coordinate

For the same example: Find the average value of y in the regionin the first quadrant bounded by y = 1− x2

To find the average y coordinate, we could use horizontal slicesand proceed the same way as before. But if you think about it, youknow the average y value of one of the vertical slices — it shouldbe the average of the y value at the top of the slice (namely1− x2) and the y value at the bottom (namely 0), which is12(1− x2). So we can proceed as before:

avg y =

ˆ 1

0

1

2(1− x2)(1− x2)dx

ˆ 1

0(1− x2)dx

=41523

=2

5.

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For the record

Just to make sure, we’ll use horizontal slices to find the averagevalue of y in the region:

Since y = 1− x2, we have x =√

1− y on the curve, so the area ofa horizontal slice is

√1− y dy . So the total number of “y points”

in a slice is y√

1− y dy . And so:

avg y =

ˆ 1

0y√

1− y dy

ˆ 1

0

√1− y dy

=41523

=2

5.

(To do both integrals, use the u-substitution u = 1− y).

The point (avg x , avg y) is called the centroid of the region.

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Taking variable density into account

In some problems, the density of the material depends on avariable in a predictable way, and we can take it into account. Itsimpliy provides an additional weighting factor.

The preceding example with variable density: Find thex-coordinate of the center of mass of the region in the firstquadrant bounded by y = 1− x2 if the density is given byδ(x) = 1 + x . (So the region is more “crowded” as you move tothe right)

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Solving the problem

The preceding example with variable density: Find thex-coordinate of the center of mass of the region in the firstquadrant bounded by y = 1− x2 if the density is given byδ(x) = 1 + x .

First we need the total mass:

mass =

ˆ 1

0(1− x2)δ(x) dx =

ˆ 1

0(1− x2)(1 + x) dx

=

ˆ 1

01 + x − x2 − x3 dx =

11

12.

Then we add up all the “weighted x points” and divide by thetotal mass:

avg x =11112

ˆ 1

0x(1−x2)(1+x) dx =

12

11

ˆ 1

0x+x2−x3−x4 dx =

23

55

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The y -coordinate of the center of mass

Example: Now find the y -coordinate of the center of mass of theregion in the first quadrant bounded by y = 1− x2 if the density isgiven by δ(x) = 1 + x .

We can do this using vertical slices as we did before, since thecenter of mass of a vertical slice is still the average of the top andbottom y -coordinates (since the density is constant in the ydirection. Therefore:

avg y =1

mass

ˆ 1

0(avg y in slice)(height of slice)δ(x) dx

=12

11

ˆ 1

0

1

2(1− x2)(1− x2)(1 + x)dx

=12

11· 7

20=

21

55

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Maxwell-Boltzmann

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Maxwell-Boltzmann

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Maxwell-Boltzmann

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Maxwell-Boltzmann

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Maxwell-Boltzmann

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