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October 15th, 2017, 11:07 AM   #1
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Spherical Challenge....

An object occupies the region in the first octant bounded by the cones
$\displaystyle \phi = \frac { \pi }{4} $ and $\displaystyle \phi = arctan 2 $,
and the sphere $\displaystyle \rho = \sqrt {6} $, and has density proportional to the distance from the origin. Find the mass.

is the following correct?

$\displaystyle
\int_{0}^{ \frac { \pi } {2} } \int_{ \frac { \pi } {4} }^{arctan(2)} \int_{0}^{ \sqrt{6} } \rho ~ \rho^2 sin \phi ~d \rho ~d \phi ~d \theta
$
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October 15th, 2017, 11:27 AM   #2
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Quote:
Originally Posted by zollen View Post
An object occupies the region in the first octant bounded by the cones
$\displaystyle \phi = \frac { \pi }{4} $ and $\displaystyle \phi = arctan 2 $,
and the sphere $\displaystyle \rho = \sqrt {6} $, and has density proportional to the distance from the origin. Find the mass.

is the following correct?

$\displaystyle
\int_{0}^{ \frac { \pi } {2} } \int_{ \frac { \pi } {4} }^{arctan(2)} \int_{0}^{ \sqrt{6} } \rho ~ \rho^2 sin \phi ~d \rho ~d \phi ~d \theta
$
yep that looks correct, though you should include a proportionality constant for completeness.
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October 15th, 2017, 12:44 PM   #3
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That would be correct if the density were equal to the distance from the origin. Instead it is "proportional" to that distance. The density is $\kappa \rho$ where $\kappa$ is the "constant of proportionality".
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Last edited by Country Boy; October 15th, 2017 at 12:46 PM.
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