Calculus 3 Timed Quiz 7

Each point in three dimensions is _______________ represented in cylindrical coordinates by using and The conversion formulas from rectangular coordinates to cylindrical coordinates are

and conversely,

A triple integral can sometimes be evaluated by transforming to cylindrical coordinates if the region of integration is -simple and the projection of onto the -plane is a region that can be described more naturally in terms of polar coordinates over the region of integration

Proposition (Triple Integral in Cylindrical Coordinates) Let be a region with upper surface and lower surface and let be the projection of the solid onto the -plane expressed in polar coordinates. Then, if is continuous on the triple integral of over is given by

Each point in three dimensions is ___________________ represented in spherical coordinates by using and The conversion formulas from rectangular coordinates to spherical coordinates are

and conversely,

Using Jacobians, we can show that the _____________ of _______________ in spherical coordinates is

Using this formula, we can form partitions and take a limit of a Riemann sums as the partitions are refined to obtain the following integral.

Proposition (Triple Integral in Spherical Coordinates) If is _____________________ on the closed bounded region , then the triple integral of over is given by

where is the region expressed in spherical coordinates.

Example (Triple Integral in Spherical Coordinates) Using spherical coordinates verify that the volume of a sphere is where is the radius of the sphere.

Solution. Since the equation of the sphere is for and Then the volume is given by the triple integral in spherical coordinates,

_____________________________

_____________________________

= _______________________________

Cite this as:
Calculus 3 Timed Quiz 7
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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