# Spherical Challenge....

#### zollen

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
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
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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• 1 person