a. Let E be the solid that lies inside the sphere x^2 + y^2 + z^2 \leq 1 and above the cone ...

Question:

(a) Let {eq}E {/eq} be the solid that lies inside the sphere

{eq}x^2 + y^2 + z^2 \leq 1 {/eq} and above the cone

{eq}z = \sqrt{3(x^2 + y^2)} . {/eq} Describe the corresponding region {eq}E' {/eq} in spherical coordinates, with the aid of inequalities.

(b) Write the volume of the solid in Part (a) as a triple integral in spherical coordinates, and evaluate the integral.

Iterated Integrals- Spherical Coordinates:

The region of integration {eq}E {/eq} is bounded by a sphere and a cone, both of which are easy to describe with spherical coordinates, so it is a good idea to try to evaluate the volume integral using spherical coordinates.

We recall that the main equations for spherical coordinates are:

{eq}\begin{align*} x&=\rho\cos\theta\sin\phi \\ y&=\rho\sin\theta\sin\phi \\ z&=\rho\cos\phi \\ \rho^2&=x^2+y^2+z^2, \end{align*} {/eq}

and the volume element is given by {eq}\rho^2\sin\phi\,d\rho\,d\theta\,d\phi. {/eq}

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For Part (a) we observe region of integration {eq}E {/eq} is bounded by a sphere and a cone, and for the volume the integrand is the constant... Cylindrical & Spherical Coordinates: Definition, Equations & Examples

from

Chapter 13 / Lesson 10
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Cylindrical coordinates and spherical coordinates are similar to the cartesian coordinate system but applied to specifying points in a three-dimensional plane. Learn how polar coordinates apply to these calculations and their use in celestial maps.