To apply a double integral to a situation with circular symmetry, it is often convenient to use a double integral in polar coordinates. We can apply these double integrals over a polar rectangular region or a general polar region, using an iterated integral similar to those used with rectangular double integrals.
The area
in polar coordinates becomes
Use
and
to convert an integral in rectangular coordinates to an integral in polar coordinates.
Use
and
to convert an integral in polar coordinates to an integral in rectangular coordinates, if needed.
To find the volume in polar coordinates bounded above by a surface
over a region on the
-plane, use a double integral in polar coordinates.
Key equations
Double integral over a polar rectangular region
Double integral over a general polar region
In the following exercises, express the region
in polar coordinates.
is the region of the disk of radius
centered at the origin that lies in the first quadrant.
In the following exercises, the integrals have been converted to polar coordinates. Verify that the identities are true and choose the easiest way to evaluate the integrals, in rectangular or polar coordinates.
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