Essential Concepts
- Several physical applications of the definite integral are common in engineering and physics.
- Definite integrals can be used to determine the mass of an object if its density function is known.
- Work can also be calculated from integrating a force function, or when counteracting the force of gravity, as in a pumping problem.
- Definite integrals can also be used to calculate the force exerted on an object submerged in a liquid.
Key Equations
- Mass of a one-dimensional object
[latex]m={\displaystyle\int }_{a}^{b}\rho (x)dx[/latex] - Mass of a circular object
[latex]m={\displaystyle\int }_{0}^{r}2\pi x\rho (x)dx[/latex] - Work done on an object
[latex]W={\displaystyle\int }_{a}^{b}F(x)dx[/latex] - Hydrostatic force on a plate
[latex]F={\displaystyle\int }_{a}^{b}\rho w(x)s(x)dx[/latex]
Glossary
- density function
- a density function describes how mass is distributed throughout an object; it can be a linear density, expressed in terms of mass per unit length; an area density, expressed in terms of mass per unit area; or a volume density, expressed in terms of mass per unit volume; weight-density is also used to describe weight (rather than mass) per unit volume
- Hooke’s law
- this law states that the force required to compress (or elongate) a spring is proportional to the distance the spring has been compressed (or stretched) from equilibrium; in other words, [latex]F=kx,[/latex] where [latex]k[/latex] is a constant
- hydrostatic pressure
- the pressure exerted by water on a submerged object
- work
- the amount of energy it takes to move an object; in physics, when a force is constant, work is expressed as the product of force and distance
