Using spherical coordinates set up a triple integral for the volume of the solid that lies within...

Question:

Using spherical coordinates set up a triple integral for the volume of the solid that lies within the sphere {eq}x^2 + y^2 + z^2= 16 {/eq}, above the {eq}xy {/eq}-plane, and below the cone {eq}z = \sqrt{x^2 + y^2} {/eq}.

Volume as Triple Integral in Spherical Coordinates

The volume of a solid region S is given by the triple integral {eq}\displaystyle \iiint_{S}\ dV, {/eq} and evaluated as an iterated triple integral by describing the region of integration S in Cartesian, cylindrical or spherical coordinates.

If the region involves spheres or cones, we can use the spherical coordinates {eq}\displaystyle \rho - \text{ the spherical radius}, \theta - \text{ the polar angle and } \varphi - \text{ the altitude angle}. {/eq}

With the description obtained in spherical coordinates, the volume integral is now converted to an iterative integral by using the following conversion for the volume differential {eq}\displaystyle dV=\rho^2 \sin\varphi d\rho d\varphi d\theta. {/eq}

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To set up the triple integral in spherical coordinates for the volume of the solid region S inside the sphere {eq}\displaystyle ... Cylindrical & Spherical Coordinates: Definition, Equations & Examples

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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.