Integral calculus
      The definite integral
      The area between the graph of a function and the x-axis over a closed interval
         The upper sum and the lower sum
      The definition of the definite integral
The area between the graph of a function and the x-axis over a closed interval
The problem of finding the area enclosed by the graph of a function and the x-axis over a closed interval is solved by use of the quadrature. The quadrature is the method of the calculation of planar areas that consists of the construction of a square with the same area as a given figure or surface.
Let y = f (x) be a continuous positive function defined on the closed interval [a, b].
Define the area A of a region of the coordinate plane bounded above by the graph of  f, below by the interval [a, b] on the x-axis, and bounded to the left by the vertical line x = a and on the right by the vertical line x = b.
The upper sum and the lower sum
Divide the given interval [a, b] into n subintervals by a finite sequence of points such that
a = x0 < x1 < x2 < . . . < xn - 1 < xn = b.
The obtained subintervals are    [a, x1], [x1, x2] . . . , [xi - 1, xi] . . . , [xn - 1, b]
 of lengths  Dx1, Dx2 . . . Dxi . . . ,   Dxn  where,  Dxi  = xi - xi - 1  and  ( i = 1, 2 , . . . , n ).
Since, by the assumption f is continuous on [a, b] then there exist the highest value M and the lowest value m of the function inside the interval [a, b] and similarly, inside the subinterval Dx1 it will have some highest value M1 and a lowest m1, in Dx2 values M2 and m2, . . . and in Dxn a function values Mn and mn.
Such that always
mi < Mi,  and  Mi < M as  mi > m
( i = 1, 2 , . . . , n ), because the highest value in a subinterval cannot exceed the highest value of the function in the whole interval [a, b], as the lowest value in a subinterval cannot be lower than that in the whole interval, as is shown in the right figure.
Now let write
       S = M1 Dx1 + M2 Dx2   Mn Dxn 
and   s = m1 Dx1 + m2 Dx2   mn Dxn,  where S represents the sum of areas of the circumscribed rectangles, each base of witch is one of the subintervals Dxi in the sequence, with height of the highest function value Mi in that subinterval. Therefore, S denotes the upper sum of  f over the interval [a, b].
While, the sum s denotes the lower sum of areas of the inscribed rectangles with the same bases and with heights that equal the lowest function value mi in each subinterval, as shows the figure above.
Since  mi < Mi  ( i = 1, 2 , . . . , n ),  then   s < S 
 and since,  m < mi  and   Mi < M  then,
  m (b - a) = m (Dx1   + Dxn) < s   and   S < M (Dx1   + Dxn) = M (b - a),
that is, the lower sum does not exceed the upper sum, both sums are bounded from above and from below.
If we now repeat the process and continue to divide each subinterval Dx into smaller intervals, so that the number of subintervals increases to infinity (n oo), that is, bases of the rectangles become smaller and smaller while the number of rectangles increases to infinity.
By passing to the infinitely small (infinitesimal) interval Dx, each of two infinite sequences of numbers S and s converge to a fixed limit value.
That is, the sequence of areas of circumscribed rectangles will decrease toward the fixed limit value, as the number of intervals increases, and the sequence of areas of inscribed rectangles will increase toward that fixed limit value.
Thus, if the sequence of circumscribed rectangles S and the sequence of inscribed rectangles s tends to the same limit value I independent of the partition of the interval [a, b], provided the length of each subinterval tends to zero while number of the partition points tend to infinity, then the limit value I equals the area A.
Therefore, using the above notation we can write
The definition of the definite integral
Finally, we say a function f is integrable on an interval [a, b] if there exists a unique number A such that
for any partition of [a, b].
If  y = f (x) is integrable on [a, b] then we call A the definite integral of f on [a, b] and write
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).
Calculus contents E
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