Learning Goal:

To understand the relationship between the step and impulse functions and use the sifting property of the impulse function to calculate the integral of a function that is the product of some function and the impulse function.

The impulse function, $($t$)$$($t$),$ is defined by the following two equations:

$$$$K$($t$)$dt$=$K$$K$($t$)$dt$=$K

$($t$)=0$$($t$)=0$ for t$0$t$0$

The first of these equations tells us that the area under the impulse function is a constant; the second equation tells us that the impulse is zero everywhere except where the argument of the function is zero. An impulse occurring at t$=$at$=$a is written $($t$$a$)$$($t$$a$).$