,
where R is a circular region. In this case the description of R in terms of
rectangular coordinates is rather complicated but R is easily described using
polar coordinates.
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Suppose that we want to evaluate a double integral
,
where R is a circular region. In this case the description of R in terms of
rectangular coordinates is rather complicated but R is easily described using
polar coordinates.
Pictures of R where polar would work (see page 1023)
Recall that we use the following equations to convert from rectangular coordinates to polar coordinates:
The region in polar coordinates has as a special name: a polar rectangle. This polar rectangle is defined below:
This region is shown graphically below:
an example of R with parts labeled. See Figure three on 1024
In order to compute the double integral
,
where R is a polar rectangle, we divide the interval
![$[a,b]$](16.4 Double Integrals in Polar Coordinates__8.png)
into m
subintervals.![$[r_{i-1},r_{i}]$](16.4 Double Integrals in Polar Coordinates__9.png)
of equal width
and we divide the interval
![$[\alpha ,\beta ]$](16.4 Double Integrals in Polar Coordinates__11.png){kind=small}
into
subintervals
[
]
of equal width
Then
the circles
and the rays
divide the polar rectangle R into the small polar rectangles shown in below:
picture with R broken down into smaller and smaller polar rectangles figure four on page 1024
The formal definition of the smaller polar rectangles are:
We then use the equation
to help us determine the area of the small rectangle. Then, we simply sum up
all of the areas of the polar rectangle and we find that the area of R to be:
Now if we start thinking of volume by making the equation
then the Riemann sum of the can be written as:
Now we take the limit of that to get the double integral in polar coordinates:
Therefore we have the following definition:
If
is continuous on a polar rectangle R given by
where
then
Do not forget to add the additional r when using polar coordinates!
Everything that we have done so far can be extended into a more complicated type of region (see picture below). Its similar to the type II rectangular regions considered in the previous section. In fact we use a formula from a previous section to obtain the following formula.
Picture of Region R on 1026 (figure 7)
If
is continuous on a polar region of the form of
Then,
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