Hence (x) dx is equivalent to the following direction a "divide b-a into n equal parts, each part being h; in (x) substitute for a successively a, a +h, a + 2h, a + (n − 1) h; add these values together, diminish h without limit." we shall have as the result ... multiply the sum by h and then If these operations are performed (b) — ↓ (ā). 5. A single term such as (x) Ax is frequently called an element. It may be observed that the limit of Σp (x) Ax will not be altered in value if we omit a finite number of its elements, or add a finite number of similar elements; for in the limit each element is indefinitely small, and a finite number of indefinitely small quantities ultimately vanishes. 6. The above process is called Integration; the quantity f(x) dx is called a definite integral, and a and b are called a ф the limits of the integral. Since the value of this definite integral is (b)(a) we must, when a function (x) is to be integrated between assigned limits, first ascertain the function(x) of which (x) is the differential coefficient. To express the connexion between (x) and (x) we have In such an equation as the last, where we have no limits assigned, we merely assert that (x) is the function from which (x) can be obtained by differentiation; called the indefinite integral of 4 (x). (x) is here 7. The problem of finding the areas of curves was one of those which gave rise to the Integral Calculus, and furnishes an illustration of the preceding articles. Let DPE be a curve of which the equation is y = (x), and suppose it required to find the area included between this curve, the axis of x, and the ordinates corresponding to the abscissæ a and b. Let OA= a, OB= b; divide the space AB into n equal intervals and draw ordinates at the points of division. Suppose OM= a + (-1) h, then the area of the parallelogram PMNp is ho {a + (r−1)h}. The sum found by assigning to r in this expression all values from 1 to n differs from the required area of the curve by the sum of all the portions similar to the triangle PQp, and as this last sum is obviously less than the greatest of the figures of which PMNQ is one, we can, by sufficiently diminishing h, obtain a result differing as little as we please from the required area. Therefore the area of the curve is the limit of the series h[p(a)+(a + h) + (a + 2h).. ${a + (n − 1) h]}, and is equal to (b) — (a). 8. If (x) be the function from which (x) springs by differentiation, we denote this by the equation [$ (x) dx = ¥ (x), and we now proceed to methods of finding (x) when (x) is given. We have shewn, Dif. Cal. Art. 102, that if two functions have the same differential coefficient with respect to a variable they can only differ by some constant quantity; hence if(x) be a function having (x) for its differential coefficient, then(x) + C, where C is any quantity independent of X, is the only form that can have the same differential coefficient. Hence, hereafter, when we assert that any function is the integral of a proposed function, we may if we please add to such integral any constant quantity. Integration then will for some time appear to be merely the inverse of differentiation, and we might have so defined it; we have however preferred to introduce at the beginning the notion of summation because it occurs in many of the most important applications of the subject. 9. Immediate integration. When a function is recognized to be the differential coefficient of another function we know of course the integral of the first. The following list gives the integrals of the different simple functions: 10. Integration by substitution. The process of integration is sometimes facilitated by substituting for the variable some function of a new variable. Suppose (x) the function to be integrated, and a and b the limits of the integral. It is evident that we may suppose x to be a function of a new variable z, provided that the function chosen is capable of assuming all the values of x required in the integration. Put then x=f(z), and let a' and b' be the values of z, which make f(z) or x equal to a and b respectively; thus a = f(a) and b = f(b). Now suppose that (x) is the function of which (x) is the differential coefficient, that is ☀ (x) b dif (x). dx [* $ (x) dx = 4 (b) — 4 (a) =↓ {ƒ(b)} −¥{ƒ (a')}. But by the principles of the Differential Calculus, therefore thus ¥ {ƒ (b')} − + {ƒ (@')} = [], $ {ƒ (z)} ƒ′(z) dz b α ƒ*$ (x) dx = [* $ (x) da dr. α dz provided we remember that when the former integral is taken between certain limits a and b, the latter must be taken between corresponding limits a' and b'. 11. As an example of the preceding article let da √ x √(2αx — a2) dx dz a be required. Assume x = thus 1 2 Here we have found the proposed integral by substituting for x in the manner indicated in the preceding article. This process will often simplify a proposed integral, but no rules can be given to guide the student as to the best assumption to make; this point must be left to observation and practice. The use of this formula is called "integration by parts." For example, consider fa For example, consider x cos ax dx. Since 1 d. sin ax cos ax = |