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4.4. THE FUNDAMENTAL THEOREM OF CALCULUS
Summary
In this section, we encountered the following important ideas:
• We can find the exact value of a definite integral without taking the limit of a Riemann
sum or using a familiar area formula by finding the antiderivative of the integrand, and
hence applying the Fundamental Theorem of Calculus.
• The Fundamental Theorem of Calculus says that if f is a continuous function on [a, b]
and F is an antiderivative of f , then
b
a
f (x) dx = F(b) − F(a).
Hence, if we can find an antiderivative for the integrand f , evaluating the definite
integral comes from simply computing the change in F on [a, b].
• A slightly different perspective on the FTC allows us to restate it as the Total Change
Theorem, which says that
b
a
f
′ (x) dx = f (b) − f (a),
for any continuously differentiable function f . This means that the definite integral of
the instantaneous rate of change of a function f on an interval [a, b] is equal to the
total change in the function f on [a, b].
Exercises
1. The instantaneous velocity (in meters per minute) of a moving object is given by the
function v as pictured in Figure 4.37. Assume that on the interval 0 ≤ t ≤ 4, v(t) is
given by v(t) = −
1
4 t 3 +
3
2 t 2 + 1, and that on every other interval v is piecewise linear,
as shown.
(a) Determine the exact distance traveled by the object on the time interval
0 ≤ t ≤ 4.
(b) What is the object’s average velocity on [12, 24]?
(c) At what time is the object’s acceleration greatest?
(d) Suppose that the velocity of the object is increased by a constant value c for all
values of t. What value of c will make the object’s total distance traveled on
[12, 24] be 210 meters?
4.4. THE FUNDAMENTAL THEOREM OF CALCULUS
Summary
In this section, we encountered the following important ideas:
• We can find the exact value of a definite integral without taking the limit of a Riemann
sum or using a familiar area formula by finding the antiderivative of the integrand, and
hence applying the Fundamental Theorem of Calculus.
• The Fundamental Theorem of Calculus says that if f is a continuous function on [a, b]
and F is an antiderivative of f , then
b
a
f (x) dx = F(b) − F(a).
Hence, if we can find an antiderivative for the integrand f , evaluating the definite
integral comes from simply computing the change in F on [a, b].
• A slightly different perspective on the FTC allows us to restate it as the Total Change
Theorem, which says that
b
a
f
′ (x) dx = f (b) − f (a),
for any continuously differentiable function f . This means that the definite integral of
the instantaneous rate of change of a function f on an interval [a, b] is equal to the
total change in the function f on [a, b].
Exercises
1. The instantaneous velocity (in meters per minute) of a moving object is given by the
function v as pictured in Figure 4.37. Assume that on the interval 0 ≤ t ≤ 4, v(t) is
given by v(t) = −
1
4 t 3 +
3
2 t 2 + 1, and that on every other interval v is piecewise linear,
as shown.
(a) Determine the exact distance traveled by the object on the time interval
0 ≤ t ≤ 4.
(b) What is the object’s average velocity on [12, 24]?
(c) At what time is the object’s acceleration greatest?
(d) Suppose that the velocity of the object is increased by a constant value c for all
values of t. What value of c will make the object’s total distance traveled on
[12, 24] be 210 meters?