Summary of Physical Applications

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]


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
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