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We integrate over regions in cylindrical coordinates.
The first way we will generalize polar coordinates to three dimensions is with
An ordered triple consisting of a radius, an angle, and a height can be graphed as
Coordinates of this type are called cylindrical coordinates.
Consider the point in cylindrical coordinates. What is this point when expressed in
Consider the point in -coordinates. What is this point when expressed in cylindrical
coordinates where ?
Triple integrals in cylindrical coordinates
If you want to evaluate this integral you have to change to a region defined in
-coordinates, and change to some combination of leaving you with some iterated
integral: Now consider representing a region in cylindrical coordinates and
let’s express in terms of , , and . To do this, consider the diagram below:
Here we see
Recalling that the determinate of a matrix gives the volume of a parallelepiped, we could
also deduce the correct form for by setting
Fubini Let be continuous on the region Then:
Write down a triple integral in cylindrical coordinates that will compute the volume
of a cylinder of radius and height .
Find the volume under above the quarter circle inside in the first quadrant.
In this case
So our integral is
Find the volume of the object defined as the intersection of the
cylinder and the sphere .