Applications of the Integral
This final application, that of finding the distance traveled by an object given its velocity at each moment, follows directly from the fundamental theorem of calculus. Suppose an object is moving in a straight line, and let its velocity at time t be given by the function f (t) . We saw earlier on, while working with derivatives, that the function F(t) giving the position of the object at time t is an antiderivative of the velocity function f (t) :
|F'(t) = f (t)|
Let us suppose that we know the velocity function f (t) (and the initial position of the object at t = 0 ) and want to determine the position function F(t) . To compute the distance the object travels from time 0 to time b , F(b) - F(0) , we apply the Fundamental Theorem of Calculus:
|F(b) - F(a) = f (x)dx|
So the area under the graph of a velocity function gives the distance traveled.