Chapter 4
The Definite Integral
4.1 Determining distance traveled from velocity
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• If we know the velocity of a moving body at every point in a given interval, can
we determine the distance the object has traveled on the time interval?
• How is the problem of finding distance traveled related to finding the area under a
certain curve?
• What does it mean to antidifferentiate a function and why is this process relevant
to finding distance traveled?
• If velocity is negative, how does this impact the problem of finding distance
traveled?
Introduction
In the very first section of the text, we considered a situation where a moving object had
a known position at time t. In particular, we stipulated that a tennis ball tossed into the
air had its height s (in feet) at time t (in seconds) given by s(t) = 64 − 16(t − 1) 2 . From
this starting point, we investigated the average velocity of the ball on a given interval
[a, b], computed by the difference quotient
s(b)−s(a)
b−a , and eventually found that we could
determine the exact instantaneous velocity of the ball at time t by taking the derivative of
207
The Definite Integral
4.1 Determining distance traveled from velocity
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• If we know the velocity of a moving body at every point in a given interval, can
we determine the distance the object has traveled on the time interval?
• How is the problem of finding distance traveled related to finding the area under a
certain curve?
• What does it mean to antidifferentiate a function and why is this process relevant
to finding distance traveled?
• If velocity is negative, how does this impact the problem of finding distance
traveled?
Introduction
In the very first section of the text, we considered a situation where a moving object had
a known position at time t. In particular, we stipulated that a tennis ball tossed into the
air had its height s (in feet) at time t (in seconds) given by s(t) = 64 − 16(t − 1) 2 . From
this starting point, we investigated the average velocity of the ball on a given interval
[a, b], computed by the difference quotient
s(b)−s(a)
b−a , and eventually found that we could
determine the exact instantaneous velocity of the ball at time t by taking the derivative of
207