A Region W In $\mathbb { R } ^ { 3 }$

question 191

Essay

A region W in $\mathbb { R } ^ { 3 }$ is described completely by $x \geq 0$ , $y \geq 0$ , $z \geq 0$ , and $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 4$ .(a) Describe or sketch this region. (b) Write an integral in rectangular coordinates which gives the volume of W. Do not work out this integral.(c) Write an integral in spherical coordinates which gives the volume of W. Find the volume of W using this integral.

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Definitions:

Place Theory

A theory in hearing that suggests that our perception of sound depends on where each component frequency produces vibrations along the basilar membrane of the inner ear.

Frequency Theory

A theory in the study of hearing that suggests pitch is determined by the rate at which nerve impulses are fired in response to sound vibrations.

High Pitches

Sounds that have a high frequency, typically perceived as shrill or squeaky, easily distinguishable from lower pitch sounds.

Low-Frequency

Low-frequency pertains to events, signals, or items occurring or operating at a relatively small number of cycles or occurrences per unit of time.