4.2. RIEMANN SUMS
229
we can of course compute the sum even when f takes on negative values. We know
that when f is positive on [a, b], the corresponding left Riemann sum L n estimates the
area bounded by f and the horizontal axis over the interval. For a function such as the
y = f (x)
a
b
c
d
y = f (x)
a
b
c
d
y = f (x)
A 1
A 2
A 3
a
b
c
d
Figure 4.17: At left and center, two left Riemann sums for a function f that is sometimes
negative; at right, the areas bounded by f on the interval [a, d].
one pictured in Figure 4.17, where in the first figure a left Riemann sum is being taken
with 12 subintervals over [a, d], we observe that the function is negative on the interval
b ≤ x ≤ c, and so for the four left endpoints that fall in [b, c], the terms f (x i )△x have
negative function values. This means that those four terms in the Riemann sum produce
an estimate of the opposite of the area bounded by y = f (x) and the x-axis on [b, c].
In Figure 4.17, we also see evidence that by increasing the number of rectangles used
in a Riemann sum, it appears that the approximation of the area (or the opposite of the
area) bounded by a curve appears to improve. For instance, in the middle graph, we use
24 left rectangles, and from the shaded areas, it appears that we have decreased the error
from the approximation that uses 12. When we proceed to Section 4.3, we will discuss the
natural idea of letting the number of rectangles in the sum increase without bound.
For now, it is most important for us to observe that, in general, any Riemann sum of a
continuous function f on an interval [a, b] approximates the difference between the area
that lies above the horizontal axis on [a, b] and under f and the area that lies below the
horizontal axis on [a, b] and above f . In the notation of Figure 4.17, we may say that
L 24 ≈ A 1 − A 2 + A 3 ,
where L 24 is the left Riemann sum using 24 subintervals shown in the middle graph, and
A 1 and A 3 are the areas of the regions where f is positive on the interval of interest, while
A 2 is the area of the region where f is negative. We will also call the quantity A 1 − A 2 + A 3
the net signed area bounded by f over the interval [a, d], where by the phrase “signed area”
we indicate that we are attaching a minus sign to the areas of regions that fall below the
horizontal axis.
229
we can of course compute the sum even when f takes on negative values. We know
that when f is positive on [a, b], the corresponding left Riemann sum L n estimates the
area bounded by f and the horizontal axis over the interval. For a function such as the
y = f (x)
a
b
c
d
y = f (x)
a
b
c
d
y = f (x)
A 1
A 2
A 3
a
b
c
d
Figure 4.17: At left and center, two left Riemann sums for a function f that is sometimes
negative; at right, the areas bounded by f on the interval [a, d].
one pictured in Figure 4.17, where in the first figure a left Riemann sum is being taken
with 12 subintervals over [a, d], we observe that the function is negative on the interval
b ≤ x ≤ c, and so for the four left endpoints that fall in [b, c], the terms f (x i )△x have
negative function values. This means that those four terms in the Riemann sum produce
an estimate of the opposite of the area bounded by y = f (x) and the x-axis on [b, c].
In Figure 4.17, we also see evidence that by increasing the number of rectangles used
in a Riemann sum, it appears that the approximation of the area (or the opposite of the
area) bounded by a curve appears to improve. For instance, in the middle graph, we use
24 left rectangles, and from the shaded areas, it appears that we have decreased the error
from the approximation that uses 12. When we proceed to Section 4.3, we will discuss the
natural idea of letting the number of rectangles in the sum increase without bound.
For now, it is most important for us to observe that, in general, any Riemann sum of a
continuous function f on an interval [a, b] approximates the difference between the area
that lies above the horizontal axis on [a, b] and under f and the area that lies below the
horizontal axis on [a, b] and above f . In the notation of Figure 4.17, we may say that
L 24 ≈ A 1 − A 2 + A 3 ,
where L 24 is the left Riemann sum using 24 subintervals shown in the middle graph, and
A 1 and A 3 are the areas of the regions where f is positive on the interval of interest, while
A 2 is the area of the region where f is negative. We will also call the quantity A 1 − A 2 + A 3
the net signed area bounded by f over the interval [a, d], where by the phrase “signed area”
we indicate that we are attaching a minus sign to the areas of regions that fall below the
horizontal axis.