246
4.3. THE DEFINITE INTEGRAL
⊳
Summary
In this section, we encountered the following important ideas:
• Any Riemann sum of a continuous function f on an interval [a, b] provides an estimate
of the net signed area bounded by the function and the horizontal axis on the interval.
Increasing the number of subintervals in the Riemann sum improves the accuracy of
this estimate, and letting the number of subintervals increase without bound results
in the values of the corresponding Riemann sums approaching the exact value of the
enclosed net signed area.
• When we take the just described limit of Riemann sums, we arrive at what we call
the definite integral of f over the interval [a, b]. In particular, the symbol
b
a
f (x) dx
denotes the definite integral of f over [a, b], and this quantity is defined by the equation
b
a
f (x) dx = lim
n→∞
n
i=1
f (x
∗
i )△x,
where △x =
b−a
n , x i = a + i△x (for i = 0, . . . , n), and x ∗
i satisfies x i−1 ≤ x ∗
i ≤ x i (for
i = 1, . . . , n).
• The definite integral
b
a
f (x) dx measures the exact net signed area bounded by f and
the horizontal axis on [a, b]; in addition, the value of the definite integral is related to
what we call the average value of the function on [a, b]: f AVG[a, b] =
1
b−a ·
b
a
f (x) dx. In
the setting where we consider the integral of a velocity function v,
b
a
v(t) dt measures
the exact change in position of the moving object on [a, b]; when v is nonnegative,
b
a
v(t) dt is the object’s distance traveled on [a, b].
• The definite integral is a sophisticated sum, and thus has some of the same natural
properties that finite sums have. Perhaps most important of these is how the definite
integral respects sums and constant multiples of functions, which can be summarized
by the rule
b
a
[c f (x) ± kg(x)] dx = c
b
a
f (x) dx ± k
b
a
g(x) dx
where f and g are continuous functions on [a, b] and c and k are arbitrary constants.
Exercises
1. The velocity of an object moving along an axis is given by the piecewise linear function
v that is pictured in Figure 4.29. Assume that the object is moving to the right when its
4.3. THE DEFINITE INTEGRAL
⊳
Summary
In this section, we encountered the following important ideas:
• Any Riemann sum of a continuous function f on an interval [a, b] provides an estimate
of the net signed area bounded by the function and the horizontal axis on the interval.
Increasing the number of subintervals in the Riemann sum improves the accuracy of
this estimate, and letting the number of subintervals increase without bound results
in the values of the corresponding Riemann sums approaching the exact value of the
enclosed net signed area.
• When we take the just described limit of Riemann sums, we arrive at what we call
the definite integral of f over the interval [a, b]. In particular, the symbol
b
a
f (x) dx
denotes the definite integral of f over [a, b], and this quantity is defined by the equation
b
a
f (x) dx = lim
n→∞
n
i=1
f (x
∗
i )△x,
where △x =
b−a
n , x i = a + i△x (for i = 0, . . . , n), and x ∗
i satisfies x i−1 ≤ x ∗
i ≤ x i (for
i = 1, . . . , n).
• The definite integral
b
a
f (x) dx measures the exact net signed area bounded by f and
the horizontal axis on [a, b]; in addition, the value of the definite integral is related to
what we call the average value of the function on [a, b]: f AVG[a, b] =
1
b−a ·
b
a
f (x) dx. In
the setting where we consider the integral of a velocity function v,
b
a
v(t) dt measures
the exact change in position of the moving object on [a, b]; when v is nonnegative,
b
a
v(t) dt is the object’s distance traveled on [a, b].
• The definite integral is a sophisticated sum, and thus has some of the same natural
properties that finite sums have. Perhaps most important of these is how the definite
integral respects sums and constant multiples of functions, which can be summarized
by the rule
b
a
[c f (x) ± kg(x)] dx = c
b
a
f (x) dx ± k
b
a
g(x) dx
where f and g are continuous functions on [a, b] and c and k are arbitrary constants.
Exercises
1. The velocity of an object moving along an axis is given by the piecewise linear function
v that is pictured in Figure 4.29. Assume that the object is moving to the right when its