Spherical Challenge....

Jan 2017
209
3
Toronto
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?

\(\displaystyle
\int_{0}^{ \frac { \pi } {2} } \int_{ \frac { \pi } {4} }^{arctan(2)} \int_{0}^{ \sqrt{6} } \rho ~ \rho^2 sin \phi ~d \rho ~d \phi ~d \theta
\)
 

romsek

Math Team
Sep 2015
2,969
1,676
USA
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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Country Boy

Math Team
Jan 2015
3,261
899
Alabama
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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