Advanced calculus; . afunction f(x) be computed from some point £ in each interval Ax{ andbe multiplied by Ax{, then the limit of the sum im [/(£)_*_! +/(&) Ax, + • • • + /&) ^] = £/(»)dx< (62) « = 00 FUNDAMENTAL RULES 25 when each interval becomes infinitely short and their number n be-comes infinite, is known as the definite integral of f(x) from a to b, andis designated as indicated. If y=f(x) be graphed, the sum will berepresented by the area undera broken line, and it is clearthat the limit of the sum, thatis, the integral, will be repre-sented by the area under thecurve y =f(x) and be
Advanced calculus; . afunction f(x) be computed from some point £ in each interval Ax{ andbe multiplied by Ax{, then the limit of the sum im [/(£)_*_! +/(&) Ax, + • • • + /&) ^] = £/(»)dx< (62) « = 00 FUNDAMENTAL RULES 25 when each interval becomes infinitely short and their number n be-comes infinite, is known as the definite integral of f(x) from a to b, andis designated as indicated. If y=f(x) be graphed, the sum will berepresented by the area undera broken line, and it is clearthat the limit of the sum, thatis, the integral, will be repre-sented by the area under thecurve y =f(x) and betweenthe ordinates x = a and x = the definite integral, de-fined arithmetically by (62),may be connected with a geo-metric concept which can serve to suggest properties of the integralmuch as the interpretation of the derivative as the slope of the tan-gent served as a useful geometric representation of the arithmeticaldefinition (2). Eor instance, if a, b, c are successive values of x, then. £ /(*) <**+X /(*)(h=L a*)dx (63) is the equivalent of the fact that the area from a to c is equal to thesum of the areas from a to b and b to c. Again, if \x be consideredpositive when x moves from a to b, it must be considered negativewhen x moves from b to a and hence from (62) $J(x)dx = -fj{x)dx. (64) Finally, if M be the maximum of f(x) in the interval, the area underthe curve will be less than that under the line y = M through thehighest point of the curve; and if m be the minimum of fix), thearea under the curve is greater than that under y = m. Hence (b - a) <£f(x)dx < M(b - a). (65) There is, then, some intermediate value m < /x < M such that the inte-gral is equal to fx(b — a); and if the line y = fx cuts the curve in apoint whose abscissa is £ intermediate between a and b, then £f(x)dx = M,(b- «) = (b - a)f(i). (65) This is the fundamental Theorem of the Mean for definite integrals. 26 INTKODUCTOKY EEVIEW The definition (62) may be applied
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