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| Integral
calculus |
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The
definite and indefinite integrals |
The area between the graph
of a function and the x-axis over a closed interval |
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Geometric interpretation of the definite integral |
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Properties of the definite integral |
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The definite integral
over interval of zero length |
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Reverse
order of integration |
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The constant multiple rule |
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The integral of the sum or difference of two functions |
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Internal addition of the definite integral |
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The definite integral of an odd function |
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The definite integral of an even function |
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The definite integral of a nonnegative and nonpositive functions |
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Relationship of the definite integrals of two functions over the same
interval of integration |
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Absolute integrability |
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The mean value theorem |
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The average value of a function over the given interval |
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| The area between the graph
of a function and the x-axis over a closed interval |
| Geometric
interpretation of the definite integral |
| Until
now we assume that the integrand, the function that is integrated, to be
nonnegative or
f(x) >
0
for all |
| x
in an interval [a,
b]. |
| Now suppose f(x)
<
0
in
the whole interval or in some of its parts then, the areas of regions
between the |
| graph
of f
and the x-axis,
which lie below or above the x-axis,
differ in the sign of f(x). |
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|
| Therefore,
the definite integral yields the algebraic sum of these areas taking
regions below the x-axis |
| negative,
as show the figures above. |
| So,
if the graph of f looks
as in the left figure above then |
 |
| hence
the definite integral represents the algebraic sum of the areas above
and below the x-axis. |
| Thus,
as the right figure above shows, follows that |
 |
|
as the area A2,
lying under the arc of the sinusoid in the interval [p,
2p]
is congruent to the area A1
in |
| [0,
p]
but with opposite sign. |
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| Properties
of the definite integral |
| 1)
The definite integral
over interval of zero length |
| Since
the definite integral we evaluate as the limit of Riemann sums, the
basic properties of limits hold for |
| integrals
as well. |
| Thus,
the limit of Riemann sums show
the first property |
 |
| as
the upper limit b
tends to lower limit a,
written b
®
a. |
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| 2)
Reverse
order of integration |
| By
reversing the upper and lower limits of integration to b
< a, that is, passing
through the x-axis
from a
to b |
| in
opposite direction, each difference, x1
-
a, x2 -
x1, . . . , b -
xn
- 1
of Riemann sums becomes |
| negative
while function values, f(xi')
can stay unchanged, therefore |
 |
| The definite integral
changes sign if the limits of integration interchange. |
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| 3)
The constant multiple rule |
| Suppose
f(x)
is integrable over the
interval [a,
b] and c
is any real number, then c f(x)
is integrable over |
| [a,
b] such that |
 |
| If
integrand is multiplied by a constant, we can factor out the constant. |
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| 4)
The integral of the sum or difference of two functions |
| Suppose
f(x)
and g(x)
are integrable over the
interval [a,
b] then |
 |
| The
integral of the sum or difference of two functions is the sum or
difference of the integrals of each function. |
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| 5)
Internal addition of the definite integral |
| Suppose
f(x)
is integrable over the
interval [a,
b] and c
is a point inside the interval, i.e., a
< c < b then, the |
| additive
property of the definite integral holds |
 |
| Let
for example c
lies outside the interval such that c
< a < b, and
assume f(x)
is integrable over the
interval |
| [c,
b] then, |
 |
| Therefore,
the above rule holds for any arrangement of three points, a,
b
and c. |
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| 6)
The definite integral of an odd function |
| If
the integrand is an odd function f
(x) = -
f
(-x),
the graph of which is symmetrical about the origin, with |
| the
interval of integration [-
a,
a],
then |
 |
|
| 7)
The definite integral of an even function |
| If
the integrand is an even function f
(x) = -
f
(x), the graph of
which is symmetrical about the y-axis,
with |
| the
interval of integration [-
a,
a],
then |
 |
|
| 8)
The definite integral of a nonnegative and nonpositive functions |
| Suppose
f
(x)
is integrable over the
interval [a,
b] and f
(x) >
0 but not identically
equal to zero, then |
 |
| If
f
(x)
is integrable over the
interval [a,
b] and f
(x) <
0 but not identically
equal to zero, then |
 |
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| 9)
Relationship of the definite integrals of two functions over the same
interval of integration |
| Suppose
f
(x) and g
(x) are integrable over the
interval [a,
b] and f
(x) >
g
(x) for all x
in [a,
b], then |
 |
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| 10)
Absolute integrability |
| Suppose
f
is integrable over the
interval [a,
b], it can be shown
that | f
(x)| is also
integrable on [a,
b] |
 |
| Since,
-
|
f
(x)| <
f
(x) <
|
f
(x)|, then |
 |
| what
means the same as the above. |
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| 11)
The mean value theorem |
| Suppose f
(x)
is integrable over the
interval [a,
b] and m
and M
are minimal and maximal value of the |
| function,
that is m
<
f
(x) <
M
for all x
in [a,
b], then |
 |
| Geometric
meaning of the above inequality is that the area under the graph of f
(x) over the
interval [a,
b] is |
| contained inside
the rectangles with the same base (b
-
a)
and of the heights m
and M. |
|
Since f
(x)
is continuous in the
interval [a,
b] it |
| takes at least
one time each value between m
and M |
|
inside the interval. |
| Therefore, there is at least one
value x
inside the |
|
interval [a,
b] such that |
 |
| meaning, there
exists the rectangle with the base |
|
 |
|
| (b
-
a)
and a height f
(x)
whose area equals the area under the graph of f
(x) over the
interval [a,
b]. |
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| 12)
The average value of a function over the given interval |
| For
a continuous function f
over an
interval [a,
b], the average value
of f
(x) is defined as |
 |
| Thus,
the average value of a function f
(x) over an
interval [a,
b] is equal to some
value of the function |
| between
its minimal and maximal value inside the interval, as shows the above
figure. |
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