Question

Use spherical coordinates to find the volume of the region outside the cone and inside the sphere rho.

a) 1/3 πrho ^3
b) 1/2 πrho ^2
c) 1/4 πrho ^4
d) 4/3 πrho ^3

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Answer 1

The correct formula for the volume of the region outside the cone and inside the sphere in spherical coordinates is
`\((1)/(3) \pi \rho^3\)` (Option A).

Step-by-step explanation:

In spherical coordinates, the volume element is expressed as
`\(dV = \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi\)`, where
`\(\rho\)` is the radial distance,
`\(\phi\)` is the polar angle, and
`\(\theta\)` is the azimuthal angle. To find the volume inside the sphere and outside the cone, we need to set up the integral over the appropriate bounds. The cone is defined by the equation
`\(\tan(\phi) = (h)/(r)\),` where
`\(h\)` is the height and
`\(r\)` is the radius. The sphere is defined by
`\(\rho = \text{constant}\)`. By setting up the triple integral with the given volume element and the appropriate bounds, we arrive at
`\(\int_0^(2\pi) \int_0^{(\pi)/(4)} \int_0^\rho \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta\)` . Evaluating this integral gives the volume
`\((1)/(3) \pi \rho^3\)`, confirming Option A as the correct answer.

Understanding how to set up and evaluate triple integrals in spherical coordinates is essential in solving volume problems with complex geometric shapes. In this case, the cone's shape introduces an additional constraint in the polar angle, affecting the limits of integration. The resulting volume expression
`\((1)/(3) \pi \rho^3\)` represents the volume of the region inside the sphere and outside the cone, providing a mathematical description of the space enclosed by these surfaces.

The correct identification of the volume formula ensures accuracy in calculating and predicting the spatial characteristics of the region defined by the cone and sphere. This mathematical representation is valuable in various fields, including physics and engineering, where understanding and manipulating volumes play a crucial role in solving practical problems.