Calculus Of One Real Variable – By Pheng Kim Ving
Chapter 9: The Integral – Section 9.3: The Definite Integral

 

9.3
The Definite Integral

 

 

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1. Riemann Sums For General Continuous Functions

 

In Section 9.2 we dealt only with continuous functions that are non-negative. For a function f non-negative 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 non-negative
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:
non-negative, non‑positive, and partly positive partly 0 partly negative.

 

Lower, upper, and general Riemann sums for non-negative 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 sub-interval, which is finite and closed. By Section 1.2.2 Theorem 2.1 on every sub-interval, f
attains a minimum and a maximum on that sub-interval.

 

 

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
color and part between it and
x‑axis if any:

 

 

Lower Riemann Sum.

 

 

 

Upper Riemann Sum.

 

 

 

General Riemann Sum.

 

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
sub-intervals) including the lower and upper ones.

 

 

Remark that these definitions are the same as those for non-negative continuous functions; see Section 9.2 Definition 5.1.

 

Signs

 

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 x-axis contributes its area to, thus increases, the Riemann sum, while each rectangle below
   the x-axis contributes the negative of its area to, thus decreases, the Riemann sum.

 

For a rectangle above the x-axis, 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
   sub-interval.

 

For a rectangle below the x-axis, if its height increases, respectively decreases, then the Riemann sum decreases,
   respectively increases.

 

Signs And Rectangles

 

 

 

Geometrically a Riemann sum for f on [a, b] is equal to the areas of all the rectangles above the x-axis minus the areas of all the rectangles below the x-axis. The Riemann sum can be positive or 0 or negative.

 

 

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2. Areas For General Continuous Functions

 

Refer to Fig. 2.1. The region between y = x2 – 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 x-axis, the vertical line x = a, and the
vertical line x = b”.

 

Area For [0, 1]

 

Let's find the area A01 of the region between y = f(x) = x2 – 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 sub-interval is (1 – 0)/n = 1/n. The left endpoints are xi = 0 +
i(1/n) = i/n for i = 0, 1, 2, ..., n – 1, so they are:

 

 

Fig. 2.1

 

Finding Areas.

 

Area For [1, 3]

 

Now let's find the area A13 of the region between f and [1, 3]. See Fig. 2.1. For the calculation of Rn[1, 3] see
9.3 Calculation Of Rn[1, 3]. We have:

 

Area For [0, 3]

 

Next let's find the area A03 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 A03. For the calculation of
Rn[0, 3] see 9.3 Calculation Of Rn[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 sub-region above the x-axis minus the area of the sub-region below the x-axis.

 

Limits Of Riemann Sums And Areas Of Regions

 

What is discussed above about the area A03 is true for general continuous functions. The limit of the sequence of the
Riemann sums is equal to the areas of all the sub-regions above the x-axis minus the areas of all the sub-regions below
the x-axis. Recall that a Riemann sum is equal to the areas of all the rectangles above the x-axis minus the areas of all
the rectangles below the x-axis. 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
rectangles above x-axis minus
areas of rectangles below
x-axis.

 

Fig. 2.4

 

So limit of Riemann sums is
areas of sub-regions above
x-axis minus areas of
sub-regions below x-axis.

 

 

The limit of the sequence of the Riemann sums for f on [a, b] is equal to the areas of all the sub-regions above the xaxis minus the areas of all the sub-regions below the x-axis. The limit can be positive or 0 or negative.

 

 

To find the area of a region that contains sub-regions below the x-axis by employing the limits of Riemann sums find the
area of each sub-region separately then add all of them up together, like we did for area A03 in Fig. 2.1.

 

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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 non-negative 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 sub-interval they may or may not attain a minimum or a
maximum on that sub-interval. 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.

 

Definition 3.1 – The Definite Integral

 

 

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.

 

Terminology

 

 

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:

 

    

 

Remarks 3.1

 

 

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
should be for clear view.

 

Fig. 3.2

 

Two of the rectangles are larger still for
description.

 

Example 3.1

 

Calculate:

 

 

if it exists, directly from the definition.

 

Solution

Let y = f(x) = x2. 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 sub-interval 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

 

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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 x-axis minus the areas of all the sub-regions below the x-axis. Therefore in particular if f is non-negative 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
sub-regions above x-axis minus
areas of all sub‑regions below
x-axis.

 

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.

 

Definition 4.1 – Area

 

The area A of the region R between f and [a, b] is defined as follows:

 

 

is the areas of all the sub-regions of R lying above the x-axis minus the areas of all the sub-regions of R lying below the x-axis. These areas are defined in parts i and ii. To find A we can find the areas of the sub-regions 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.

 

Example 4.1

 

Use geometry to find:

 

 

Solution

Let f(x) = y = x – 1. Refer to Fig. 4.2. Clearly the x-intercept is x = 1. We have f(–2) = – 2 – 1 = –3 and f(5) = 5 – 1 =
4. The area A1 of the sub-region over f under [–2, 1] is A1 = (1/2)(1 – (–2))| f(–2)| = (3/2)|–3| = 9/2 square units. The
area A2 of the sub-region under f over [1, 5] is A2 = (1/2)(5 – 1) f(5) = 2(4) = 8 square units. Thus:

 

 

Fig. 4.2

 

EOS

 

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5. Integrability Of Continuous Functions

 

Every continuous function is integrable.

 

Theorem 5.1

 

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.

 

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6. Properties Of The Definite Integral

 

Theorem 6.1

 

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

 

 

Example 6.1

 

Given that:

 

 

Solution

EOS

 

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7. Differential Calculus And Integral Calculus

 

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.

 

 

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Problems & Solutions

 

 

Solution

 

The length of each sub-interval is (b – 0)/n = b/n. The endpoints xi's are xi = 0 + i(b/n) = i(b/n) for i = 0, 1, 2, ..., n,
so they are:

 

 

 

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2. Find the area A of a rectangle with base b and height h using the definite integral.

 

Solution

 

 

Set up a coordinate system such that the rectangle's lower base is on the x-axis with its lower left corner at the origin
and hence its left height is on the y-axis. Then A is the area of the region under y = f(x) = h over [0, b]. Use the right
endpoints xi for i = 1, 2, …, n. The length of each sub-interval is (b – 0)/n = b/n. As f(xi) = h for all i and f is continuous
we have:

 

 

Note

 

That's exactly the same as is obtained by the geometric formula A = base x height = bh.

 

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3. Evaluate the definite integral:

 

   

 

    by interpreting it as area.

 

Solution

 

 

 

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4.  Consider the regular partition of order 2 of [0, 2].

 

   

 

Solution

 

a.

 

 

b.

 

 

c.

 

 

d.

 

 

 

5. Given:

 

   

 

Solution

 

 

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6.  Suppose f and g are continuous functions, f is even, g is odd, and a is a constant. Prove that:

 

   

 

Solution

 

 

 

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