Cylindrical coordinates assigned an ordered triple to points in space. If the rectangular coordinates of the point are , then are the polar coordinates of the point ; the third coordinate z is the same in both cylindrical and rectangular.
Example. A point has rectangular coordinates . Find its cylindrical coordinates.
Since the point lies above the second quadrant, I've rotated the coordinates axes so that the second quadrant is "in front". From the picture, it's clear that . By Pythagoras, .
The cylindrical coordinates are .
Example. A point has cylindrical coordinates . Find its rectangular coordinates.
The point lies below the fourth quadrant. Its rectangular coordinates are .
Example. Find the equation of the unit sphere in cylindrical coordinates.
Setting , I get
Example. Convert to cylindrical coordinates.
Example. What is the set of points which satisfy the cylindrical coordinate equation ?
The equation is , so or . The locus consists of two concentric cylinders of radius 1 and 2 having the z-axis as their axis.
To convert a triple integral to cylindrical coordinates:
1. Convert to cylindrical coordinates using the polar coordinate conversion equations.
2. To obtain the limits of integration, describe the region R by inequalities in cylindrical coordinates.
3. Replace with .
In converting a double integral from rectangular to polar, we replace with , so this is reasonable from that point of view. You can see the factor of r from the change of variables formula. The transformation is
The Jacobian is
Assuming that r is positive, the change of variables formula tells us to replace with .
Example. Let R be the region bounded below by and above by . Compute
The intersection of and is , a circle of radius 6 centered at the origin. Hence, the region of integration is
Note that . The integral is
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