Question

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

, where \({\rm{B}}\)is the unit ball

\({{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{ + }}{{\rm{z}}^{\rm{2}}} \le {\rm{1}}\)

Short answer

The required triple integral is

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

Consider the given integral and simplify,

Because the sphere's interior is symmetrical, the integral of an odd function is 0

As a result, the red part is 0

Where \({\rm{V}}\)is the sphere's volume

\(\begin{array}{c}{\rm{3 \times }}\frac{{\rm{4}}}{{\rm{3}}}{\rm{\pi }}{{\rm{r}}^{\rm{3}}}\\{\rm{ = 4\pi }}{{\rm{r}}^{\rm{3}}}\\{\rm{ = 4\pi }}\end{array}\)

Hence, the required triple integral is