CHAPTER 8 Further Applications of Integration
Concepts and Skills
In this chapter, we will review many ways that definite integrals can be used to solve a variety of
problems, notably distance traveled by an object in motion along a line. We’ll see that in a variety
of settings accumulated change can be expressed as a Riemann sum whose limit becomes an
integral of the rate of change.
For BC students, we’ll expand our discussion of motion to include objects in motion in a plane
along a parametrically defined curve.A. MOTION ALONG A STRAIGHT LINE
If the motion of a particle P along a straight line is given by the equation s = F(t), where s is the
distance at time t of P from a fixed point on the line, then the velocity and acceleration of P at time t
are given respectively by
This topic was discussed as an application of differentiation. Here we will apply integration to find
velocity from acceleration and distance from velocity.
Distance
If we know that particle P has velocity v(t), where v is a continuous function, then the distance
traveled by the particle during the time interval from t = a to t = b is the definite integral of its speed:
If v(t) 0 for all t on [a, b] (i.e., P moves only in the positive direction), then (1) is equivalent to
similarly, if v(t) 0 on [a, b] (P moves only in the negative direction), then (1) yields
If v(t) changes sign on [a, b] (i.e., the direction of motion changes), then (1) gives the total
distance traveled. Suppose, for example, that the situation is as follows:
Then the total distance traveled during the time interval from t = a to t = b is exactly