Question

Question: Evaluate the triple integral using only geometric interpretation and symmetry.

, where \({\rm{C}}\)is the cylindrical region

\({{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}} \le {\rm{4, - 2}} \le {\rm{z}} \le {\rm{2}}\)

Short answer

The required solution is \({\rm{64\pi }}\).

Step-by-step solution

Concept Introduction

A multiple integral is a definite integral of a function of many real variables in mathematics (particularly multivariable calculus), such as f(x, y) or f(x, y) (x, y, z). Integrals of a two-variable function over a region

Explanation of the solution

1) We begin by rewriting the integral in order to more quickly identify the application of geometric interpretation or symmetry.

2) If we note that the {function in the second integral is odd with respect to \({\rm{y}}\)and \({\rm{C}}\)is symmetrical over the \({\rm{xz}}\) plane.} we may considerably decrease our effort. As a result, we can deduce that the second integral's value is 0. Let's start with the first.

3) Let us now recall the following property:

4) Because our cylinder has a height of 4 and a radius of 2, we can compute its volume as follows:

\(\begin{array}{c}{\rm{V(E) = \pi \times (2}}{{\rm{)}}^{\rm{2}}}{\rm{*4}}\\{\rm{ = 16\pi }}\end{array}\)

5) And here's what we've come up with:

\(\begin{array}{c}{\rm{ = 4 \times V(E)}}\\{\rm{ = 4 \times 16\pi }}\\{\rm{ = 64\pi }}\end{array}\)

Hence, the required solution is \({\rm{64\pi }}\).