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4.4. THE FUNDAMENTAL THEOREM OF CALCULUS
• For a moving object with instantaneous velocity v(t), the object’s change in position
on the time interval [a, b] is given by
b
a
v(t) dt, and whenever v(t) ≥ 0 on [a, b],
b
a
v(t) dt tells us the total distance traveled by the object on [a, b].
• For any continuous function f , its definite integral
b
a
f (x) dx represents the total
net signed area bounded by y = f (x) and the x-axis on [a, b], where regions that lie
below the x-axis have a minus sign associated with their area.
• The value of a definite integral is linked to the average value of a function: for a
continuous function f on [a, b], its average value f AVG[a, b] is given by
f AVG[a, b] =
1
b − a
b
a
f (x) dx.
The Fundamental Theorem of Calculus now enables us to evaluate exactly (without taking a
limit of Riemann sums) any definite integral for which we are able to find an antiderivative
of the integrand.
A slight change in notational perspective allows us to gain even more insight into
the meaning of the definite integral. To begin, recall Equation (4.4), where we wrote the
Fundamental Theorem of Calculus for a velocity function v with antiderivative V as
V (b) − V (a) =
b
a
v(t) dt.
If we instead replace V with s (which represents position) and replace v with s ′ (since
velocity is the derivative of position), Equation (4.4) equivalently reads
s(b) − s(a) =
b
a
s
′ (t) dt.
(4.5)
In words, this version of the FTC tells us that the total change in the object’s position
function on a particular interval is given by the definite integral of the position function’s
derivative over that interval.
Of course, this result is not limited to only the setting of position and velocity. Writing
the result in terms of a more general function f , we have the Total Change Theorem.
The Total Change Theorem: If f is a continuously differentiable function on [a, b]
with derivative f ′ , then f (b) − f (a) =
b
a
f ′ (x) dx. That is, the definite integral of
the derivative of a function on [a, b] is the total change of the function itself on [a, b].
The Total Change Theorem tells us more about the relationship between the graph of
a function and that of its derivative. Recall Figure 1.18, which provided one of the first
times we saw that heights on the graph of the derivative function come from slopes on the
graph of the function itself. That observation occurred in the context where we knew f
4.4. THE FUNDAMENTAL THEOREM OF CALCULUS
• For a moving object with instantaneous velocity v(t), the object’s change in position
on the time interval [a, b] is given by
b
a
v(t) dt, and whenever v(t) ≥ 0 on [a, b],
b
a
v(t) dt tells us the total distance traveled by the object on [a, b].
• For any continuous function f , its definite integral
b
a
f (x) dx represents the total
net signed area bounded by y = f (x) and the x-axis on [a, b], where regions that lie
below the x-axis have a minus sign associated with their area.
• The value of a definite integral is linked to the average value of a function: for a
continuous function f on [a, b], its average value f AVG[a, b] is given by
f AVG[a, b] =
1
b − a
b
a
f (x) dx.
The Fundamental Theorem of Calculus now enables us to evaluate exactly (without taking a
limit of Riemann sums) any definite integral for which we are able to find an antiderivative
of the integrand.
A slight change in notational perspective allows us to gain even more insight into
the meaning of the definite integral. To begin, recall Equation (4.4), where we wrote the
Fundamental Theorem of Calculus for a velocity function v with antiderivative V as
V (b) − V (a) =
b
a
v(t) dt.
If we instead replace V with s (which represents position) and replace v with s ′ (since
velocity is the derivative of position), Equation (4.4) equivalently reads
s(b) − s(a) =
b
a
s
′ (t) dt.
(4.5)
In words, this version of the FTC tells us that the total change in the object’s position
function on a particular interval is given by the definite integral of the position function’s
derivative over that interval.
Of course, this result is not limited to only the setting of position and velocity. Writing
the result in terms of a more general function f , we have the Total Change Theorem.
The Total Change Theorem: If f is a continuously differentiable function on [a, b]
with derivative f ′ , then f (b) − f (a) =
b
a
f ′ (x) dx. That is, the definite integral of
the derivative of a function on [a, b] is the total change of the function itself on [a, b].
The Total Change Theorem tells us more about the relationship between the graph of
a function and that of its derivative. Recall Figure 1.18, which provided one of the first
times we saw that heights on the graph of the derivative function come from slopes on the
graph of the function itself. That observation occurred in the context where we knew f