Lecture Notes, M261-004, Double Integrals in Polar Form
Oct 20, 2008
We have learned about double integrals in rectangular coordinates. We can think of this as
dividing the region up into small rectangles, calculating the volume of each block given by these
small rectangles the height of the function, and then adding these volumes up. Today, we consider
a different approach to integration where instead of dividing up the region into rectangles, we
divide it up intowedges of circles. This is integration in polar coordinates and it often makes
integrands easier to evaluate. This is one example of change of variables – or substitution – in
multiple integrals. We look at the following topics.
• Review of Polar Coordinates
• dA in Polar Coordinates
• Finding Limits of Integration in Polar Coordinates
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Review of Polar Coordinates
In rectangular coordinates, we identify a point by its distance from the origin along the x-axis and
its distance from the origin along the y-axis. (We could also think of the coordiantes as the lengths
of the projections of the vector from the origin to the point onto the x and y axes respectively.)
There are many other ways we could identify a point, however, and each one would give rise to an
alternate coordinate system.
The polar coordinate system is one of the most useful of these alternate coordinate systems. It
identifies a point by the angle that the vector from the origin to the point makes with the x-axis
and the distance from the origin to the point, (the length of the vector from the origin to the
point.) We can go between rectangular and polar coordinates by
x = r cos θ
y = r sin θ
and by
p
x2 + y 2
y
θ = tan−1
x
r=
2
dA in Polar Coordinates
When we form an integral in polar coordinates, it looks like
Z Z
Z θ 1 Z r1
f (r, θ)rdrdθ
f (r, θ)dA =
R
θ0
1
r0
You may have noticed that we have picked up an extra r in the integrand. To help understand where
this r comes from, we will consider the expression dA in both rectangluar and polar coordinates.
In rectangular coordiantes, we have
dA = dxdy
If we integrate this,
Z Z
dxdy = (y1 − y0 )(x1 − x0 )
we get the area of a small rectangle. When we integrate dA in polar coordinates, we want to get
the area of a small wedge of a circle. And thus,
Z
Z Z
θ
1
rdrdθ = r2 dθ = r2
2
2
This is the desired area.
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Finding Limits of Integration in Polar Coordinates
The last issue is finding limits of integration for a region. This is done the same way as in
rectangluar coordiantes–we sketch the region and the find the limits for r and θ from the sketch
we have drawn. We can convert an integral in rectangular coordinates by finding the correct limits
for the region and then taking
Z Z
Z Z
f (x, y)dxdy =
f (r cos θ, r sin θ)rdrdθ
R
R
We can get all the same interpretations of an integral in polar coordinates that we get for an
integral in rectangular coordinates, including volume, area, average value, etc... Polar coordinates
just give an easier way to evaluate the integral in some cases.
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Examples
Example 1. Change the integral to polar coordinates and evaluate it.
Z 1Z
dydx
√
−1
− 1−x2
Example 2. Change the integral to polar coordinates and evaluate it.
Z Z √ 2
4−y
2
(x2 + y 2 )dydx
0
0
Example 3. Change the integral to polar coordinates and evaluate it.
Z
0
1
Z
√
1−x2
2
e−(x
+y 2 )
dxdy
0
Example 4. Find the area of the region enclosed by the positive x-axis and spiral r = 4θ/3, 0 ≤
θ ≤ 2π.
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