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Mathematical Expression Editor

A planimeter computes the area of a region by tracing the boundary.

Green’s Theorem may seem rather abstract, but as we will see, it is a fantastic tool
for computing the areas of arbitrary bounded regions. In particular, Green’s Theorem
is a theoretical planimeter. A planimeter is a “device” used for measuring the area
of a region. Ideally, one would “trace” the border of a region, and the planimeter
would tell you the area of the region.

How is Green’s Theorem a planimeter?

Recall Green’s Theorem:

Green’s Theorem If the components of have continuous
partial derivatives and is a boundary of a closed region and parameterizes in a
counterclockwise direction with the interior on the left, then

Given a vector field , if , then the left-hand side of the conclusion of Green’s Theorem
gives the area of the region :

So now the question becomes, which vector fields have ? Here are three basic
candidates:

The key idea that connects the three vector fields above is:

Their curl is .They
are conservative fields.They are gradient fields.When used in combination with
Green’s Theorem, they help compute area.

Once we have a vector field whose curl is , we may then apply Green’s Theorem to
use a line integral to compute the area.

You must parameterize with for such that:

is drawn in a counterclockwise direction.

is drawn exactly once.

The interior of is to the left of the direction of .

Computing areas with Green’s Theorem

Now let’s do some examples.

Compute the area of the trapezoid below using Green’s Theorem.

In this case, set . Since , Green’s Theorem says: We need to parameterize our paths
in a counterclockwise direction. We’ll break it into four line segments each
parameterized as runs from to :

where:
and each draws the line as runs from to . Write:
For each of the integrands above, say , we will write
and combine them into a single integral. Write with me
So this is the area of the trapezoid.

Compute the area of the ellipse using Green’s Theorem.

To start, we’ll set . Since ,
Green’s Theorem says: We can parameterize the boundary of the ellipse with
for . Write with me
Done. Green’s Theorem for the win!

Finally, what do you do if you have a very strangely shaped curve? You approximate
it with a polygonal curve. Check out next example.

Compute the area of the polygonal region below using Green’s Theorem.

In this case, set . Now we’ll break this curve into six line segments that
draw this curve in a counterclockwise fashion as a parameter runs from to .

where:
and each draws the line as runs from to . Write:
For each of the integrands above, say , we will write
and combine them into a single integral. Write with me
So this is the area of the region.

Green’s Theorem gives a fairly easy method for computing any the area of any
polygonal region. Any region with a “smooth” border can be approximated by a
polygonal region. The upshot? Green’s Theorem is a powerful tool for computing
area.

The shoelace algorithm

Green’s Theorem can also be used to derive a simple (yet powerful!) algorithm (often
called the “shoelace” algorithm) for computing areas. Here’s the idea: Suppose you
have a two-dimensional polygon, where the vertices are identified by their
-coordinates:

Here we see a polygon with vertices, and the “dashed-line” means that there could
be more to this polygon than “meets the eye.” So to compute the area, here is a neat
trick. Write all of the coordinates in a column, writing the starting coordinate twice,
both at the beginning and at the end:

Now multiply entries of the columns diagonally down from the left to the right and
add them together

to obtain: Now, multiply entries of the columns diagonally down from the left to the
right and subtract them from our previous sum

to obtain:
The area of the polygon in question will be: The algorithm is called the “shoelace”
algorithm because of the crisscrossing pattern you see above.

Compute the area of the following polygon:

The area is square units.

Why does the shoelace algorithm work?

Now we are going explain why the shoelace algorithm works via Green’s Theorem.
The restrained young mathematician may protest that we are using a “crane to crush
a fly,” but whatever. We like Green’s Theorem.

Shoelace Given a polygon with vertices at we may compute the area of the
polygonal region by setting and computing:

To start, recall that if , then . Hence
Green’s Theorem states: This means we can compute the area of the region , by
evaluating the line integral on the right along the polygonal boundary . Since we’re
supposing that the vertices of the polygon are can be broken into edges, (if a
polygon has vertices, it has edges). We can parameterize each edge as
with . Moreover
this is just saying that the line integral along the perimeter of the polygon
is the sum of the line integrals along the edges. Now write with me
So now we may write
Noting that the terms and will cancel with each other as we cycle through the sum,
we find that
Since this value relies on being parameterized in a counterclockwise fashion, we take
the absolute value to ensure a correct answer (just in case the young geometer
accidentally parameterized in a clockwise fashion). Thus we have completed the
explanation of the shoelace algorithm.

A student is working with a pentagon:

and using the shoelace algorithm:

computes the area of the pentagon as:
So the student concludes that the area is Is this correct?