An introduction to the summation of differences of a function; an elementary exposition of the nature of the algebraic processes replaced by the abbreviations of the infinitesimal calculus, by B. F. Groat.

CHAPTER III

AREAS AND LIMITS OF SUMS
22. WE proceed to examine a method for finding the area
enclosed by a given curve, the axis of x, and any two ordinates; and thence a method for finding the limit of a sum of
n7 terms of a certain form, when n = oo.
Let y=f(x) be the equation of the curve, f(x) being an
increasing function in the first quadrant. Suppose we require
the area enclosed by the curve, the ordinates f(a), f(b), and
the x axis. Suppose b>a. Then the base of the area is
(b - a) in length. Divide the base into n equal parts and erect
ordinates at the points of division. There will be (n + I) of
these ordinates, YP y2,... Yn+l, and their common distance apart
will be ba, which call Ax-h. Further, we can calculate
n
the lengths of these ordinates from the equations yl =f(a),
Y2 =f(a + ), y,, = f(a + I - I ) =/f(b - - J), y,+l = f(b).
If from the points of intersection of these ordinates with
the curve, parallels to x be drawn in the positive direction to
the next ordinate in each case, we shall form a system of n
rectangles,
Y1h, 'j,2,... y.A,
whose area, yl/ +y2/1 + * +y,,,
s.less than the required area; each rectangle lying on the
positive side of its generating ordinate. If the parallels are
drawn in the negative direction, the area, y2/z + Y3ai +. + y,+lh,
will be greater than the required area, and each of the n
rectangles will lie on the negative side of its generating ordinate.
If f(x) be a decreasing function, the inequalities will be
inverted. If f(x) be part of the time increasing and part of
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