Q. 21 - 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 ρ(x, y,z) is a density function in grams per cubic centimeter on the three-dimensional region , give the units of the expression. 

${\iiint }_{\mathrm{\Omega }} x\rho (x,y,z)dV$

Step-by-Step Solution

Answer

Quantity: First momentum

The unit for momentum is grams centimeter.

Step 1. Given information.

Given:

${\iiint }_{\mathrm{\Omega }} x\rho (x,y,z)dV$

We have to identify the quantities determined by the given integral expression.

Step 2. Identify the quantity

The objective of the problem is to describe about the triple integral ${\iiint }_{\mathrm{\Omega }} x\rho (x,y,z)dv$ and find its units.

By the definition of the first momentum M of the solid $\mathrm{\Omega }$ the triple integral first

momentum.

Hence the given relation represents the first momentum M of the solid $\mathrm{\Omega }$

Since if $\rho$ represents the density of the solid $\mathrm{\Omega }$ then ${\iiint }_{\mathrm{\Omega }} x\rho (x,y,z)dv$represents the first momentum.

The unit for momentum is grams centimeter.