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 previously as an application of differentiation. Here we
will apply integration to find velocity from acceleration and distance from
velocity.
DistanceIf 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)