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4.4. THE FUNDAMENTAL THEOREM OF CALCULUS
4.4 The Fundamental Theorem of Calculus
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How can we find the exact value of a definite integral without taking the limit of a
Riemann sum?
• What is the statement of the Fundamental Theorem of Calculus, and how do
antiderivatives of functions play a key role in applying the theorem?
• What is the meaning of the definite integral of a rate of change in contexts other
than when the rate of change represents velocity?
Introduction
Much of our work in Chapter 4 has been motivated by the velocity-distance problem: if
we know the instantaneous velocity function, v(t), for a moving object on a given time
interval [a, b], can we determine its exact distance traveled on [a, b]? In the vast majority
of our discussion in Sections 4.1-4.3, we have focused on the fact that this distance traveled
is connected to the area bounded by y = v(t) and the t-axis on [a, b]. In particular, for
any nonnegative velocity function y = v(t) on [a, b], we know that the exact area bounded
by the velocity curve and the t-axis on the interval tells us the total distance traveled,
which is also the value of the definite integral
b
a
v(t) dt. In the situation where velocity is
sometimes negative, the total area bounded by the velocity function still tells us distance
traveled, while the net signed area that the function bounds tells us the object’s change in
position. Recall, for instance, the introduction to Section 4.2, where we observed that for
the velocity function in Figure 4.31, the total distance D traveled by the moving object on
[a, b] is
D = A 1 + A 2 + A 3 ,
while the total change in the object’s position on [a, b] is
s(b) − s(a) = A 1 − A 2 + A 3 .
While the areas A 1 , A 2 , and A 3 , which are each given by definite integrals, may be
computed through limits of Riemann sums (and in select special circumstances through
familiar geometric formulas), in the present section we turn our attention to an alternate
approach, similar to the one we encountered in Activity 4.2. To explore these ideas further,
we consider the following preview activity.
Preview Activity 4.4. A student with a third floor dormitory window 32 feet off the
ground tosses a water balloon straight up in the air with an initial velocity of 16 feet
4.4. THE FUNDAMENTAL THEOREM OF CALCULUS
4.4 The Fundamental Theorem of Calculus
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How can we find the exact value of a definite integral without taking the limit of a
Riemann sum?
• What is the statement of the Fundamental Theorem of Calculus, and how do
antiderivatives of functions play a key role in applying the theorem?
• What is the meaning of the definite integral of a rate of change in contexts other
than when the rate of change represents velocity?
Introduction
Much of our work in Chapter 4 has been motivated by the velocity-distance problem: if
we know the instantaneous velocity function, v(t), for a moving object on a given time
interval [a, b], can we determine its exact distance traveled on [a, b]? In the vast majority
of our discussion in Sections 4.1-4.3, we have focused on the fact that this distance traveled
is connected to the area bounded by y = v(t) and the t-axis on [a, b]. In particular, for
any nonnegative velocity function y = v(t) on [a, b], we know that the exact area bounded
by the velocity curve and the t-axis on the interval tells us the total distance traveled,
which is also the value of the definite integral
b
a
v(t) dt. In the situation where velocity is
sometimes negative, the total area bounded by the velocity function still tells us distance
traveled, while the net signed area that the function bounds tells us the object’s change in
position. Recall, for instance, the introduction to Section 4.2, where we observed that for
the velocity function in Figure 4.31, the total distance D traveled by the moving object on
[a, b] is
D = A 1 + A 2 + A 3 ,
while the total change in the object’s position on [a, b] is
s(b) − s(a) = A 1 − A 2 + A 3 .
While the areas A 1 , A 2 , and A 3 , which are each given by definite integrals, may be
computed through limits of Riemann sums (and in select special circumstances through
familiar geometric formulas), in the present section we turn our attention to an alternate
approach, similar to the one we encountered in Activity 4.2. To explore these ideas further,
we consider the following preview activity.
Preview Activity 4.4. A student with a third floor dormitory window 32 feet off the
ground tosses a water balloon straight up in the air with an initial velocity of 16 feet