Calculus Of One Real Variable –
By Pheng Kim Ving 
9.3 
Return
To Contents
Go To Problems & Solutions
1. Riemann Sums For General Continuous Functions 
In Section
9.2 we dealt only with continuous functions that are nonnegative. For a
function f
nonnegative and
continuous on [a, b] the limit of any sequence
of Riemann sums is equal to the area of the region under the graph of f
over [a, b]. We now consider continuous
functions that are general in terms of sign, ie ones that are either
nonnegative
or non‑positive or partly positive partly 0 partly negative.
Again by general continuous functions we mean continuous functions
that include all 3 possibilities in terms of sign:
nonnegative, non‑positive, and partly positive partly 0 partly negative.
Lower, upper, and general Riemann sums for nonnegative
continuous functions are defined in Section
9.2 Part 5. Now
we define them for general continuous functions. Let f be a general continuous function on [a, b]. See Figs. 1.1, 1.2, and
1.3. Let n be an
arbitrary positive integer and set up a regular partition of order n of [a, b].
Since f is
continuous on [a,
b] it's continuous on every
subinterval, which is finite and closed. By Section
1.2.2 Theorem 2.1 on every subinterval, f
attains a minimum and a maximum on that subinterval.
Fig. 1.1
Lower Riemann Sum. 
Fig. 1.2
Upper Riemann Sum. 
Fig. 1.3
General Riemann Sum. 
Fig. 1.4
Comprises part with indicated

Definitions 1.1 – Riemann Sums
For a function f continuous on [a, b]:
Note that a general Riemann sum of order n may be any sum of order n (choosing any set of
points of the 
Remark that these definitions are the same as those for nonnegative continuous functions; see Section 9.2 Definition 5.1.
Refer to Fig. 1.5. Consider the general Riemann sum:
Fig. 1.5
General Riemann Sum. 
Remarks On The Rectangles
For a given regular partition:
– Each rectangle above the xaxis
contributes its area to, thus increases, the Riemann sum, while each rectangle
below
the xaxis contributes
the negative of its area to, thus decreases, the Riemann sum.
– For
a rectangle above the xaxis, if its height increases,
respectively decreases, then the Riemann sum increases,
respectively decreases. The height of
a rectangle can be changed by selecting different points in the corresponding
subinterval.
– For
a rectangle below the xaxis, if its height increases,
respectively decreases, then the Riemann sum decreases,
respectively increases.
Geometrically a Riemann sum for f on [a, b] is equal to the areas of all the rectangles above the xaxis minus the areas of all the rectangles below the xaxis. The Riemann sum can be positive or 0 or negative. 
Go To Problems & Solutions Return To Top Of Page
2. Areas For General Continuous Functions 
Refer to Fig. 2.1. The region between y = x^{2} – 1 and [0, 3]
is the colored region. When we for simplicity say “the region
between f and [a, b]” we mean “the plane region bounded by the
graph of f, the xaxis, the vertical line x = a, and the
vertical line x = b”.
Let's find the area A_{01} of the region
between y = f(x) = x^{2} – 1 and [0,
1]. See Fig. 2.1. Utilize a regular partition of order
n of [0, 1] and the left
endpoints. The length of each subinterval is (1 – 0)/n = 1/n.
The left endpoints are x_{i}
= 0 +
i(1/n) = i/n for i = 0, 1, 2, ..., n – 1, so they are:
Fig. 2.1
Finding Areas. 
Now let's find the area A_{13} of the region
between f and [1,
3]. See Fig. 2.1. For the calculation of R_{n}[1,
3] see
9.3
Calculation Of R_{n}[1,
3]. We have:
Next let's find the area A_{03} of the region between f and [0, 3]. See Fig. 2.1. Clearly:
Let's find the limit of the sequence of the Riemann sums for
f on [0, 3] and compare it to A_{03}. For the calculation of
R_{n}[0,
3] see 9.3
Calculation Of R_{n}[0,
3]. We have:
The limit of the sequence of the Riemann sums isn't equal to
the area of the region between f
and [0, 3] but it's equal to
the area of the subregion above the xaxis
minus the area of the subregion below the xaxis.
What is discussed above about the area A_{03} is true for general continuous functions. The limit of the
sequence of the
Riemann sums is equal to the areas of all the subregions above the xaxis minus the areas of all
the subregions below
the xaxis.
Recall that a Riemann sum is equal to the areas of all the rectangles above the
xaxis minus the areas of all
the rectangles below the xaxis.
So the relationship between the limit of Riemann sums and the area of a region
is similar
to that between Riemann sums and the areas of the rectangles. This is plausible
because the area of a region is the limit
of the sequence of the areas of the rectangles. The relationship is carried
along “to the limits”. See Figs. 2.2, 2.3, and
2.4.
Fig. 2.2 Relationship is carried along “to the limits”. 
Fig. 2.3
Riemann sum is areas of 
Fig. 2.4
So limit of Riemann sums is 
The limit of the sequence of the Riemann sums for f on [a, b] is equal to the areas of all the subregions above the x‑axis minus the areas of all the subregions below the xaxis. The limit can be positive or 0 or negative. 
To find the area of a region that contains subregions below
the xaxis by
employing the limits of Riemann sums find the
area of each subregion separately then add all of them up together, like we
did for area A_{03} in Fig. 2.1.
Go To Problems & Solutions Return To Top Of Page
3. The Definite Integral 
The functions we've been dealing with since Section
9.2 are continuous. For them the concept of the area of the region
between their graphs and a closed finite interval in their domains is
clearly meaningful. It can be shown that for them the
lower and upper Riemann sums have the same limit; see theorem
5.1 (this limit may or may not be equal to the area, as
discussed earlier in this section). There are discontinuous functions on a closed
finite interval for which the concept of
such an area is still meaningful. For them the lower and upper Riemann sums
have the same limit. There are
discontinuous functions on a closed finite interval for which the
concept of such an area has lost all meaning. For them the
lower and upper Riemann sums don't have the same limit. So it's reasonable to
accept that the concept of such an area is
meaningful iff the lower and upper Riemann sums have the same limit.
The concept of the definite integral of functions of 1 real
variable is based on the concept of area. In particular the
definite integral of nonnegative f
over [a, b] is to be equal to the area
of the region under f
over [a, b]. We saw above
that the concept of such an area is meaningful iff the lower and upper Riemann
sums have the same limit (note that a
sequence that approaches infinity has no limit; thus if 2 sequences have the
same limit then that limit must be finite).
As a consequence the concept of the definite integral requires that the lower
and upper Riemann sums have the same
limit. If this common limit exists, the area may or may not be equal to it,
however the definite integral is defined to be it.
For functions that are discontinuous but bounded, on each
subinterval they may or may not attain a minimum or a
maximum on that subinterval. Thus the definitions of lower and upper Riemann
sums, as presented in Definitions 1.1,
have to be modified to include both functions that are continuous and ones that
are discontinuous and bounded. This is
usually done in a more advanced calculus course. However the definition of the
definite integral is practically the same as
for continuous functions.
and the function f is said to be integrable on [a, b]. 
Geometrically the definite integral is the common limit at
infinity of the sequence of sums of signed areas of lower
rectangles and of the sequence of sums of signed areas of upper rectangles if
this common limit exists.
ii. a and b are called limits of integration; a is the lower limit and b the upper limit.
iii. f is
called the integrand.
iv. x is called the variable of integration.
v. Recall that dx is the differential
of x;
see Section
4.3 Definitions 2.1. If there are 2 or more variables, the differential
tells which variable the definite
integral is taken with respect to. For example:
ii. As:
for arbitrary positive integers m and n.
Fig. 3.1
Infinitely many infinitesimally thin rectangles with
infinitesimally small widths dx
and signed heights f(x). Rectangles in picture
are larger than they 
Fig. 3.2
Two of the rectangles are larger still for 
Calculate:
if it exists, directly from the definition.
Let y = f(x) = x^{2}. Let n
be an arbitrary positive integer and set up the regular partition of order n of [1, 4]. See Figs. 3.3
and 3.4. The length of each subinterval is (4 – 1)/n = 3/n.
The endpoints are:
Fig. 3.3
Lower Riemann Sum.

Fig. 3.4
Upper Riemann Sum.

See Fig. 3.4. The calculation of the upper Riemann sum is carried out in 9.3 Calculation. We have
EOS
Go To Problems & Solutions Return To Top Of Page
4. Areas 
As the definite integral of a function f on [a,
b] is the common limit of the
lower and upper Riemann sums it may or may
not be equal to the area of the region between f and [a,
b].
The definite integral of f on [a, b] is equal to the areas of all the sub‑regions above the xaxis minus the areas of all the subregions below the xaxis. Therefore in particular if f is nonnegative on [a, b] then the definite integral is equal to the area of the entire region, which is under f over [a, b]. 
For example, see Fig. 4.1:
Fig. 4.1
Definite integral is areas of all 
Areas of some familiar geometric regions like triangles or
rectangles or trapezoids have been defined. For example the
area of a rectangle is base times height. We’ve not yet defined areas of
regions with curved boundaries. Now we're going
to do just that. We'll use the definite integral to define areas of general
regions including ones with curved boundaries.
The area A of the region R between f and [a, b] is defined as follows:

is the areas of all the subregions of R lying above the xaxis minus the areas of all the subregions of R lying below the xaxis. These areas are defined in parts i and ii. To find A we can find the areas of the subregions individually and add them up together to get A. 
This definition is general in the sense that it applies to
familiar geometric plane regions as well as regions with curved
boundaries. For an example of a familiar geometric plane region see Problem & Solution 2.
We see that areas of rectangles are used to define Riemann
sums, which in turn are used to define the definite integral,
which in turn is used to define areas of general regions. Let's call areas of
rectangles as “particular areas” and those of
general regions as “general areas”. Then we get this chain of concepts:

Remark that general areas include particular ones.
Use geometry to find:
Solution
Let f(x) = y = x
– 1. Refer to Fig. 4.2. Clearly the xintercept
is x = 1. We
have f(–2) = – 2
– 1 = –3 and f(5) = 5 –
1 =
4. The area A_{1} of the
subregion over f
under [–2, 1] is A_{1} = (1/2)(1 –
(–2)) f(–2) = (3/2)–3 = 9/2 square units.
The
area A_{2} of the
subregion under f
over [1, 5] is A_{2} = (1/2)(5 – 1)
f(5) = 2(4) = 8 square units.
Thus:
Fig. 4.2

EOS
Go To Problems & Solutions Return To top Of Page
5. Integrability Of Continuous Functions 
Every continuous function is integrable.
If f is continuous on [a, b] then it's integrable there and:

Although this theorem is straightforward its proof involves
subtle properties of the real number system derived from its
completeness property and thus is beyond the scope of this tutorial and
consequently is omitted.
By this theorem it's sufficient to use only any 1 type of
Riemann sum to find the definite integral of a function continuous
on a closed finite interval by using the definition.
Go To Problems & Solutions Return To top Of Page
6. Properties Of The Definite Integral 
Suppose functions f and g are continuous on an interval containing the points a, b, and c, where a < b and c can be anywhere in the interval, and let k be a constant. By theorem 5.1 f and g are integrable on any closed interval with these points as endpoints. Then:

Proof
i. Since f + g
is continuous on [a, b], by theorem 5.1 it's integrable there, and:
ii. Similar to part i.
iii. This follows from the fact that all the Riemann
sums are 0, which in turn follows from the fact that all the rectangles in all
the Riemann sums have bases of
length 0.
iv. We have:
v. We have:
which is equivalent to the inequality to be proved.
EOP
Remark 6.1
Fig. 6.1

Given that:
EOS
Go To Problems & Solutions Return To top Of Page
Differential calculus is
concerned with the derivative and integral calculus is concerned with the
integral. Following is a
comparison between the derivative and the integral of functions of 1 real
variable.
Problems & Solutions 
Solution
The length of each subinterval is (b – 0)/n
= b/n. The endpoints x_{i}'s are x_{i} = 0
+ i(b/n) = i(b/n) for i
= 0, 1, 2, ..., n,
so they are:
2. Find the area A of a rectangle with base b and height h using the definite integral.
Set up a coordinate system such that the rectangle's lower
base is on the xaxis
with its lower left corner at the origin
and hence its left height is on the yaxis.
Then A is the
area of the region under y
= f(x) = h over [0, b].
Use the right
endpoints x_{i} for
i = 1, 2, …, n. The length of each
subinterval is (b –
0)/n = b/n. As f(x_{i}) = h
for all i and f is continuous
we have:
That's exactly the same as is obtained by the geometric formula A = base x height = bh.
3. Evaluate the definite integral:
by interpreting it as area.
Solution
4. Consider the regular partition of order 2 of [0, 2].
Solution
a.
b.
c.
d.
5. Given:
Solution
6. Suppose f and g are continuous functions, f is even, g is odd, and a is a constant. Prove that:
Solution
Return To Top Of Page Return To Contents