Applications of Integration

The area between the graphs of two functions is equal to the integral of a function, $f(x)$, minus the integral of the other function, $g(x)$: $A = \int_a^{b} ( f(x) - g(x) ) \, dx$.

Volumes of complicated shapes can be calculated using integral calculus if a formula exists for the shape's boundary.

The average of a function $f(x)$ over an interval $[a,b]$ is $\bar f = \frac{1}{b-a} \int_a^b f(x) \ dx$.

In the shell method, a function is rotated around an axis and modeled by an infinite number of cylindrical shells, all infinitely thin.

Forces may do work on a system. Work done by a force ($F$) along a trajectory ($C$) is given as $\int_C \mathbf{F} \cdot d\mathbf{x}$.

Disc and shell methods of integration can be used to find the volume of a solid produced by revolution.