Q. 22 - 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. 

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

Step-by-Step Solution

Answer

Quantity: Centre of mass

No unit.

Step 1. Given information.

Given:

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

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

Step 2. Identify the quantity

By definition 13.19, $\overline{z}=\frac{M_{xy}}{m}$ represents the third ordinate z of centre of mass of $\mathrm{\Omega }$

Hence the given relation represents the third ordinate z of centre of mass of

$\mathrm{\Omega }$

Thus, There are no units for centre of mass.