. Applied calculus; principles and applications . c ^ c If OC = c, and CPs is the corresponding ordinate, then,area APPiB = area APP3C - area BPiPsC;and hence rf(x)dx^ rf(x)dx- r s{x)dx = f f W dx + Pf (x) dx, by Art. 131. *J a *J c Note. — It may be seen that Jf(x)dx= I f(a — x) dx, for each is F(a) — F(o). Thus, - Cfia -x)d{a-x)= -F{a- x)T = F{a)-F (0) = / f{x)dx, 133. Mean Value of a Function. — The mean value of/ {x) between the values / (a) and / (6) is tJ a dx b — a Let area APPiB represent the definite integral I / (x) dx. 212Then J f (x) dx = INTEGRAL CALCULUS areaAPPi5 = area of a rec
. Applied calculus; principles and applications . c ^ c If OC = c, and CPs is the corresponding ordinate, then,area APPiB = area APP3C - area BPiPsC;and hence rf(x)dx^ rf(x)dx- r s{x)dx = f f W dx + Pf (x) dx, by Art. 131. *J a *J c Note. — It may be seen that Jf(x)dx= I f(a — x) dx, for each is F(a) — F(o). Thus, - Cfia -x)d{a-x)= -F{a- x)T = F{a)-F (0) = / f{x)dx, 133. Mean Value of a Function. — The mean value of/ {x) between the values / (a) and / (6) is tJ a dx b — a Let area APPiB represent the definite integral I / (x) dx. 212Then J f (x) dx = INTEGRAL CALCULUS areaAPPi5 = area of a rectangle with base AB and height greater than AP but less than BPi= AB. CPi= (6 — a)f{c), where OC = c. Hence f(c) x) dx where/ (c) is the mean value off (x) for values of x that varycontinuously from a to The mean value may be defined to be the height of arectangle which has a base equal to 6 — a and an area equi-valent to the value of the integral. Example 1. — To find the mean value of the function Vxfrom X = 1 to X = 4:. Let OPPi be the locus of y f(c) Vx, OA- = 1 and OB = 4. £■ Ux _i3 >- 1] = 14 4- 1 9 = 1| = = CP, mean value,c = X = (-V^)2 = -W- = = OC. = - = EVALUATION OF DEFINITE INTEGRALS 213 Example 2. — To find the mean value of sin 0 as 0 variesfrom 0 to 7r/2, or from 0 to tt. I sinBde -cos<9 7r/2-0 ^ V2 ^2 TT I sinddd — cos 0 ^^^^^ ^r- = J-0 = f = TT — U TT X Example 3. — To find the mean length of the ordinates ofa semi-circle of radius a, the ordinates for equidistant in-tervals on the arc. £ sinddd —acos^ ^ = - J^ = — = a. TT — 0 TT Example 4. — To find the mean length of the ordinates ofa semi-circle of radius a, the ordinates for equidistant inter-vals along the diameter. r Va - ^ ^^ . „ _ . J-a 1 ^/-7
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