render it easily intelligible by full detail, and by the solution of several problems. In order that the student may find in the volume all that he requires, a large collection of examples for exercise has been appended to the different chapters. These examples have been selected from the College and University Examination Papers, and have been carefully verified, so that it is hoped that few errors will be found among them. The short introduction to the Integral Calculus which is given at the end of my treatise on the Differential Calculus, has been incorporated in the present work so as to render it complete in itself. The student has been occasionally referred to the Differential Calculus for such information as he would require before commencing the study of the Integral Calculus. I. TODHUNTER. ST JOHN'S COLLEGE, March 1857. INTEGRAL CALCULUS. CHAPTER I. MEANING OF INTEGRATION. EXAMPLES. 1. IN the Differential Calculus we have a system of rules by means of which we deduce from any given function a second function called the differential coefficient of the former; in the Integral Calculus we have to return from the differential coefficient to the function from which it was deduced. We do not say that this is the object of the Integral Calculus, for the fundamental problem of the subject is to effect the summation of a certain infinite series of indefinitely small terms; but for the solution of this problem we must generally know the function of which a given function is the differential coefficient. This we now proceed to shew. 2. Let (x) denote any function of x which remains finite and continuous for all values of x comprised between two fixed values a and b. Let x1, x,... x2-1 be a series of values lying between a and b, so that a, x1, x2, ...X-1, b are in order of magnitude ascending or descending. We propose then to find the limit of the series (x, − a) $ (a) + (x2 − x ̧) $ (x ̧) + (x3 — X„) ¤ (X2) ................ + (b − Xn-1) $ (xn−1), when x-a, x-x1, ... b-x are all diminished without limit, and consequently n increased without limit. Put x1— a = h1, x12 − x1 = h2, ... b − x-1= h2; thus the series may be written h ̧¢ (a) +h2$ (x1)...... + hn-1 $ (Xn-2) + hn & (xn−1), T. I. C. 2 1 and may be denoted by Zho (x), for it is the sum of a number of terms of which ho(x) may be taken as the type. Since each of the terms of which h is the type may be considered as the difference between two values successively ascribed to the variable x, we may also use the symbol (x) Ax as the type of the terms to be summed, and Σp (x) Ax for the sum. ... We may shew at once that p(x) Ax can never exceed a certain finite quantity. For let A denote the numerically greatest value which (x) can have when x lies between a and b; then (x) Ax is numerically less than (h,+h2+ +h) A, that is less than (ba) A. We now proceed to determine the limit of Σo (x) Ax. Let (x) be such a function of x that (x) is the differential coefficient of it with respect to x. Then we know that the (x + h) — 4(x) limit of h (x). Hence we may put where P1, P2, when h is indefinitely diminished is ↓ (x1) — (a) = h1{$(a) +p1}, † (x) — y (x,) = h2{$ (x,) + P2}, † (xn−1) — † (xn−2) = hn_1{Þ (xn_2) + Pr-1}, ... Pn ultimately vanish. From these equations we have by addition ψ (6) - ψ (α) = Σφ (α) Δα + Σηρ. Now Σhp is less than (ba) p' where p' denotes the greatest of the quantities P1, P2, ... P; hence Zho ultimately vanishes, and we obtain this result-the limit of Σ(x) Ax when each of the quantities of which Ax is the type diminishes indefinitely is (b)(a). 3. The notation used to express the preceding result is the symbol is an abbreviation of the word "sum," and dx represents the Ax of Σo (x) Ax. |