Q. 16 - Calculus

Question

Hydrostatic force is the product of weight-density, area, and depth. When we integrate to find accumulated hydrostatic force, we integrate the product of weight-density, length, and depth. Express the previous two sentences in mathematical notation. Then explain why area is used in one, but length in the other. 

Step-by-Step Solution

Answer

The mathematical notations are $F=\omega Ad and {\int }_a^b\omega dldx$.Area is used in one, but length in the other because on integrating all the lengths of the subdivided regions, we will get the area.

Step 1. Given information.

The given statements are,

Hydrostatic force is the product of weight-density, area, and depth. 

When we integrate to find accumulated hydrostatic force, we integrate the product of weight-density, length, and depth. 

Step 2. Mathematical notation.

The mathematical notations are,

$$\begin{gathered} F=\omega Ad\text{ and }{\int }_a^b\omega dldx \\ F=Force, \\ A=Area \\ d=dis\tan ce \\ \omega =weighted-density \\ l=length \end{gathered}$$

Step 3. Explanation.

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