48 CHAPTER IV. MISCELLANEOUS REMARKS. 37. WE have at the beginning of this book defined the integral of (x) between assigned limits a and b as the limit of a certain sum Σ (x) Ax, and have denoted this limit by a (x) da. We have shewn that this limit is known as soon as we know the function (x) of which (x) is the differential coefficient. In the pages immediately following we gave methods for finding (x) in different cases. We shall now add some miscellaneous remarks and theorems, some of which will recall the attention of the student to the process of summation which we placed at the foundation of the subject. 38. Suppose we wish to find the integral of sin x between limits a and b immediately from the definition. By Art. 4 we have to find the limit when n is infinite of h [sin a + sin (a + h) + sin (a + 2h)..............+ sin {a + (n − 1) h}] 39. Required the limit when n is made infinite of the Comparing this with Art. 4 we see that the required limit is n 40. We define (x) dx as the limit when ʼn is infi nite of h ̧ Þ(a) +h2Þ(x ̧)...............+ h2 & (xn−1)• Now let A and B be the greatest and least values which (x) takes between the limits a and b; then the series is less than (h1 + h2+ + h2) A, T. I. C. 4 The limit must therefore be equal to (b-a) C where C is some quantity lying between A and B; but since (x) is supposed continuous, it must, while x ranges from a to b, pass through every value between A and B, and must therefore be equal to when x has some value between a and b. Thus C=(a+0 (b− a)} where is some proper fraction, and [ ̊ $ (x) dx = (b − a) $ {a + 0 (b − a)}. 41. The truth of the equation will appear immediately; for suppose (x) to be the integral of (x), then we have on the left-hand side For putting a-x=z we have f$ (a = x) dx = - [4 (2) dz, therefore Of course ՐԱ ["4 (2) dz = ["p(x) dæ, since it is indifferent whe 0 0 ther we use the symbol x or z in obtaining a result which does not involve x or z. The second integral, by changing x into 2a-x', will be found equal to Hence, if (x)=(2a-x) for all values of x comprised between 0 and a, we have 42. Such equations as those just given should receive careful attention from the student, and he should not leave them until he recognizes their obvious and self-evident truth. cos3 e de is by definition the limit when n is infinite of the series h {cos3 h + cos3 2h + cos3 3h...... + cos3 (n − 1) h} where nhπ. Now cos3 h = cos3 (n-1) h, cos3 2h = cos3 (n-2) h, ......; thus the positive terms of the series just balance the negative terms and leave zero as the result. In the same way the truth of [" п sin Ꮎ ᏧᎾ = 2 sin Ꮎ ᏧᎾ 0 follows immediately from the definition of integration, and the fact that the sine of an angle is equal to the sine of the supplemental angle. rb 43. Suppose b greater than a and (x) always positive between the limits a and b of x; then every term in the series ≥4 (~) Ax is positive, and hence the limit (a) dæ must be a positive quantity. a 44. The statement of the last article supposes that (x) is always finite between the limits a and b; it must be remembered that this condition was expressly introduced in the fundamental proposition, Art. 2. If therefore the function to be integrated becomes infinite between the limits of integration, the rules of integration cannot be applied; at least the case must be specially examined. ra dx Lov ; the value of this integral is 45. Consider (12) 0 2-2 (1-a). Here the function to be integrated becomes infinite when x=1; but the expression 2-2/(1-a) is finite when a 1. Hence in this case we may write =2, provided that we regard this as an abbrevia |