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You might think that when computing areas there are a lot of formula that you need

to memorize. Well, fortunately, that's not the case.

There's one formula that you need to know for area.

And that is that area is the integral of the area element dA.

What do I mean by that? Let's recall some of the classical area

formulae that you learned when you were in school.

The area of a rectangle, or a parallelogram, or a triangle, or a

circular disk. These you know, but do you know why

they're true? They're true because they are The

integral of the corresponding area element.

Let's see how this works in a few examples.

Recall a parallelogram. Now you would probably argue by

rearrangement that if you cut into pieces and reshuffle.

Then you can build a rectangle, and we all know the area of a rectangle.

Well, that's good but there's another way to think about this.

Consider your parallelogram in the x y plane with the base on the x axis and

then slice it into horizontal strips. Of, if you will, an infinitismal

thickness, dy. Then what is the width of each strip?

It is, let's call it b, the length of the base of the parallelogram.

Now, how do we get the area out of this? The area element?

The area of this thin strip is well, b times dy, thinking of it as a rectangle

with one edge of infinitesmal length. The area is the integral of the area

element that is the integral of bdy as y goes from 0 to h the height in the

parallelogram and now lets see how it falls out the integral of a constant dy

is that constant b times y. Evaluate from 0 to h, and we obtain

simply the times h which we all know is the area.

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This also helps us make sense of the area of a triangle.

You're probably used to thinking of it as half the area of the bounding rectangle

by using geometry and similar triangles. But we could do things a little bit

differently. If we also arrange this triangle so that

the base is along the X axis and shear it horizontally.

So that it becomes a right triangle. We see, we have not changed the area

since we've preserved the horizontal area elements.

Now the hypotenuse has the equation of y equals h over b times x.

Now let's consider a slightly different area element.

This one, veritical The height of that vertical strip is, from our equation for

y, h/b*x, the thickness is dx, and thus, computing the area is the intergral of

the area element, and what do we get. We get the integral of h over b x dx as x

goes from 0 to b. Now, what do we see?

Well, h over b is a constant x integrated step x squared over 2.

Evaluating from 0 to b gives h b squared over 2b.

Minus 0. Simplifying we get one half base times

height. Very simple.

The example of a circular disc finaly takes us to a nonlinear case.

The classical means for determining his area was to use an Angular variable.

And to approximate as a limit of triangles.

Let us do likewise. This is perhaps not exactly how

Archimedes did it, but it's kind of close.

Consider an angular variable, theta. And consider a wedge.

Defined by an infinitesmal angle d theta. Then, thinking of this wedge as

approximately a triangle. Not exactly, but approximately a triangle

to first order, what would the dimensions be?

Well, we would say that the height of the triangle is roughly r, the radius of the

circle. The base is going to have length r times

d theta. With that in mind, the area element is

what? It is one half the base times the height,

that is r times r d theta. The area, being the integral of the area

element, is the integral of 1/2 r squared d theta, as theta sweeps all the way

around from zero to 2 pi. Now, we know what, well, wait a minute,

there's no theta in the integrand. There's just r squared, which is a

constant, as is 1/2, so we can pull out the 1/2 r squared.

The integral of d theta is, of course, theta.

Evaluating that from zero to 2 pi gives 2 pi times 1/2 r squared.

That is, pi R squared, the classical answer done very simply.

But this is not the only way to do this problem.

Consider if we used a radial variable, and we sweep out thin annuli and fill up

the disc in that manner. Let's call that radial variable t.

The annular strip is going to have thickness dt, and now we need to argue

that the area of this infinitesimal annular strip is The circumference, 2 pi

t times the thickness, d t. I'm not going to argue that that's really

correct right now. Let me just state it, and let's see what

happens when we integrate it. We get the integral of 2 pi t d t Aa t

goes from zero to r, the radius. That is, we get pi t squared as t goes

from zero to r. That again yields our answer, which, if

nothing else, tells us that our area element was indeed correct.

That is a very simple integral. But notice that it's not the only way

that we can do it. Still, we can cut the disk into slices.

Let's say vertical slices. And we sweep from left to right.

Placing the circular disk at the center of the x,y plane.

Allows us to use a strip of width dx, and height given by what?

Well, the equation of the circle is x squared plus y squared equals r squared.

Solving for y gives plus or minus square root of r squared minus x squared.

The area element, in this case, is going to be the height, that is, twice root r

squared minus x squared, times the width dx.

Then we see that integrating to get the area gives the integral of twice root r

squared minus x squared dx. Evaluating the limits from minus r to r,

what do we get? Well, this integral is not as easy as the

other two that we've done. It is an even integrand or a symmetric

domain, so could pull out the two, double it, and integrate from zero to r.

But we're still not done. We would need to use trigonometric

substitution, something of the form x equals r sine u and do a little bit of

work. I'm going to leave it to you to verify

that what one gets for that integral is pi r squared over 4.

Leading to the final answer that we all know and love.

Notice that depending on which way you split the domain up into area elements.

The resulting integral may be trivial, or may be quite involved.

This then leads to the example that we all know and love, the area between two

curves. Let's say f on the top, and g on the

bottom. Now, it's very clear given what we've

done, how to determine the area element. In this case, using a vertical strip,

it's height is going to be f(x) minus g(x).

its width is d(x). And so, integrating this, we see the

classical formula that you've no doubt seen before.

The area is the integral f or x minus g of x dx, as x goes from a to b.

One interesting application of this formula is to economics.

In the definition of something called the genie index.

This is a ratio that is used to quantify income and equality.

In a fixed population, it relies on one slightly technical definition and the

rest is simple calculus. >> Let us denote by f of x, the

fraction of the total income earned by the lowest x fraction of the populous,

the domain is x goes from 0 to 1 and you should think of x as a percentage of the

population And f(x) as the amount of income earned by that lowest x percent.

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The genie index, denoted g, quantifies how far from a flat, or even

distribution. The income is.

And what do I mean by that? Well, consider what this function, f,

looks like. It tells you how much of the net income

is being earned by what fraction of the population.

A flat income would be the diagonal line where x equals y.

Why? Well the diagonal line would mean that if

you look at the lowest 50% of the population, they earn 50% of all income.

Of course, in practice that's not true. They are in Less.

Folks at a higher income earn more and a larger percentage of the net income

earned. The Gini index is a measure of how far

from the perfectly flat, or equitable, distribution you are.

And it's defined as follows. It's the ratio of the area between the

flat distribution, y equals x, and the true income distribution, y equals f.

It's the ratio of that. And, the area between y equals x and y

equals 0. Well, that denominator is just telling

you the area of the triangle, and since the base and the height are 1, that's

1/2. So, another way to write the Gini index

is as the numerator, this area between F and x.

Namely x minus f of x, d x value of it from zero to one divided by one-half,

that is doubled. Let's look at this in a specific example.

Let's compute the gini index for a power law distribution.

Assuming that income is distributed like a polynomial, x to the n, then what would

we get? Well, we integrate x minus f(x) from 0 to

1 and double it. Well, that integral is almost trivial.

It's going to be x squared over two minus x to the end plus one over and plus one.

Evaluating from zero to one and then doubling.

What do we get? We get one minus two over and plus one.

Putting that over a common denominiator yields the result of N minus one over N

plus one. If we look at a little bit of economic

data, we find that in the year 2010, the state of New York had a GINI index of one

half approximately. If the income in New York state were

distributed according to a monomial, then what would that monomial be?

Well we see that it would be a cubic distribution.

We can't conclude that, but if it were a monomial it would be a cubic

distribution. That's a nice application.