Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 1] The Definite Integral as the Limit of a Sum 139
If the function / (x) is continuous on [a, b], it is integrable on [a, b]\ i.e.,
the limit of (2) exists and is independent of the mode of partition of the
interval of integration [a, b] into subintervals and is independent of the
choice of points in these subintervals. / Geometrically, the definite integral
(2) is the algebraic sum of the areas of the figures that make up the curvilinear
trapezoid aABb, in which the areas of the parts located above the #-axis
are plus, those below the jc-axis, minus (Fig. 37).
The definitions of integral sum and definite integral are naturally generalized
to the case of an interval [a, b], where a > b.
Example 1. Form the integral sum Sn for the function
on the interval [1,10] by dividing the interval into n equal parts and choosing
points |/ that coincide with the left end-points of the subintervals
[xit xi+l ]. What is the lim S n equal
Solution. Here, Ax. =
to?
n -+ CO
101 9 . = t
and c/ = J
-. Whence
n n
Hence (Fig. 38),
lim S n -58-L.
n ->> oo 2
Example 2. Find the area bounded by an arc of the parabola
jc-axis, and the ordinates * = 0, and x = a (a > 0).
Solution. Partition the base a into n equal y
parts = .
Choosing
the value of the func-
tion at the beginning of each subinterval, we will
have
The areas of the rectangles are obtained by mul-
tiplying each yk by the base A*= (Fig. 39).
Summing, we get the area of the step-like figure Fig. 39
Using the formula for the sum of the squares of integers,
2> n(n+\)(2n+\)
6
= x* t
the