Q. 20 - Calculus

Question

Identify the quantities determined by the integral expressions in Exercises $19–24$. If $x, y,$ and $z$are all measured in centimeters and $\rho (x, y,z)$ is a density function in grams per cubic centimeter on the three-dimensional region $\Omega$, give the units of the expression.

$\underset{\Omega }{\iiint }\rho \left(x,y,z\right)dV$.

Step-by-Step Solution

Answer

The triple integral $\underset{\Omega }{\iiint }\rho \left(x,y,z\right)dV$ represents the mass of the solid $\Omega$ and the units for mass are grams.

Step 1 . Given information

$\underset{\Omega }{\iiint }\rho \left(x,y,z\right)dV$.

Step 2 . Units for the given triple integral:

By the definition if $\rho$ represents the density of the solid, then $\underset{\Omega }{\iiint }\rho \left(x,y,z\right)dV$ represents the mass of the solid $\Omega$.

Hence the given triple integral $\underset{\Omega }{\iiint }\rho \left(x,y,z\right)dV$ represents the mass of the solid $\Omega$. The units for mass are grams.