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Integral
calculus
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The definition of the
definite integral
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| Finally,
we say a function
f
is integrable on an interval [a,
b] if there exists a
unique number A
such that |
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for
any partition of [a,
b]. |
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If
y = f (x)
is integrable on [a,
b]
then we call A
the definite integral of f
on [a,
b] and write |
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| where
∫ symbolize the sum, the function
f (x) is called
the integrand, the differential dx
shows that x
is the variable
of integration and, the numbers a
and b
are called the limits of integration (a
is the lower limit and b
is
the upper limit). |
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Riemann
sum
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| Until
now, in the definition of the sums, S
and s
we've used the maximum and the minimum values, Mi
and mi
of a given continuous function
f, so that mi
< f (x)
< Mi for x
in [xi
-
1, xi],
i = 1, 2 , . . . , n. |
| Now,
if we arbitrarily choose a point xi'
in every interval Dxi
and make products f (xi' )
Dxi,
then |
| mi
Dxi
<
f (xi' )
Dx
< Mi
Dxi,
i = 1, 2 , . . . , n and
by adding up |
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left-hand and right-hand sides of the above inequality are the sums, s
and S
respectively that, because of continuity of f,
tend to the same limit value I
when the number
of subintervals n
increases to infinity, such that the length of
every interval Dxi
tends to zero, for any partition of the interval [a,
b]
and arbitrarily chosen
points xi'
in the subintervals [xi
-
1, xi].
Hence,
the middle term |
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| called
a Rieman sum, will tend to the same limit value.
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| Therefore,
if f
is a positive continuous function on the interval [a,
b] then, the definite
integral of the function from a
to b
is defined to be the limit
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| Note
that the limit value of the sum changes as the number
of subintervals n
increases to infinity while the length
(Dxi)
of each tends to zero, for any partition of the interval [a,
b].
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Calculating
a definite integral from the definition
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As the sequence of inscribed rectangles s
tends to the definite
integral increasingly while the sequence of circumscribed rectangles S
tends to the same value decreasingly then |
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| Thus,
we
can approximate the area under the graph of a function over the interval
[a,
b]
to any desired level of accuracy using
the Riemann sum of inscribed or circumscribed rectangles. |
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area of the ith
rectangle f (xi' )
Dxi,
denoted as height times base, represents the ith
term of Riemann sum
and is called the element of area. |
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we use the partition of the interval [a,
b] into n
equal subintervals (regular partition) then |
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be
the length of the intervals [xi
-
1, xi],
i = 1, 2 , . . . , n. |
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Contents
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Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved.
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