Q. 14 - Calculus

Question

Mass is the product of density and volume. When we integrate to find accumulated mass, we integrate the product of density and cross-sectional area. Express the previous two sentences in mathematical notation. Then explain why volume is used in one, but area in the other 

Step-by-Step Solution

Answer

The mathematical notations are $m=\rho V\text{ and }{m}_i={\int }_a^b\rho Adx.$ Volume is used in one, but area in the other because on adding all the areas of the subdivided regions, we will get the volume.

Step 1. Given information.

We are given,

Mass is the product of density and volume. 

When we integrate to find accumulated mass, we integrate the product of density and cross-sectional area 

Step 2. Mathematical Notation.

We can express the given statements as,

$$\begin{gathered} m=\rho V\text{ and }{m}_i={\int }_a^b\rho Adx \\ Where m=mass, \\ \rho =density \\ V=Volume \\ A=area \end{gathered}$$

Step 3. Explanation.

We will divide the entire surface in many subparts and then integrate. Now, Density is constant, therefore, integration will not make any difference. Volume is used in one, but area in the other because on adding all the areas of the subdivided regions, we will get the volume.