CHAPTER XIX
INTEGRAL CALCULUS. DEFINITE INTEGRALS. MISCELLANEOUS APPLICATIONS
GENERAL PROPOSITIONS
1. It is desirable to place on record several general propositions, in regard to change of limits, which are in the nature of being self-evident.
In Chapter XVI it has been shown that
f bf (x) dx = F (b) — F (a), 6 where f(x) is the differential coefficient of F(x). It follows that
- I.faL2 b f (x) dx = b f () dz (1), sinc e neither x nor z occurs in the result.
- II.fa f (x) dx = —fbf (x)dx (2).
Thus the interchange of the limits results in a change of sign of the definite integral.
On the left-hand side we have regarded the increment of dx as positive, so that, while x increases from a to b, the value of the integral is F (b) — F (a).
On the right-hand side the increment dx is negative and x decreases from b to a, giving a value for the integral of
F(a)— F(b).
III.f f (x) dx =f f (x) dx +f f (x) dx (3).
abaFor the left-hand side is F(c) — F(a) and the right-hand side is F (c) — F (b) + F (b) — F (a).
- IV.(af (x) dx = f f (a — x) dx (4).
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110 INTEGRAL CALCULUS
For if we substitute a — z for x, then when x = a, z = 0, and when ,c=0,z=a.
Also — dz = dx.
Therefore r r
Cif J f(x)dx=fa-f(a—z)dz= (a—z)dz (by II)
=f f (a — x) dx (by I).
- 2.Differentiation of Definite Integrals.
bLetu = f f (x, c) dxabe a definite integral where the quantity c is independent of x, and the limits a and b are independent of c.To find c let Au be the change in u corresponding to a change Ac in c. Then, since the limits are unaltered,
Au= f b{f(x, c+Ac)—f(x, c)} dx.
Au— f{f(x,c+Ac)-f(x,c)} Therefore AcAc dx.Proceeding to the limit, we havedubd_.f(x,c) (5).do fa do dx* Thus the differential of the definite integral is reached by a process of differentiating under the sign of integration.
- 3.An important use of this theorem is that of finding the values of other integrals from those of known form. For example, if the equation
f e axdx—1 oa
* The student is referred to more advanced treatises for exceptions to this general result.
MISCELLANEOUS APPLICATIONS 111
be differentiated n times with respect to a, we get
xn e—ax dx = an+i .o
- 4.Areas of Curves. It has been shown that geometrically the definite integral
b
f f (x) dx represents the area enclosed between the curve y = f (x)
. aand the axis of x bounded by the ordinates x = a and x = b.
Example 1. Prove that the area of the parabola y' = 4ax bounded by the curve, the axis of x and any ordinate is two-thirds of the rectangle conby the ordinate and the intercept on the axis of x.
Let OW = b.r[ Then area OPN= f bydx =J b \/4ax dx = 2a1 [ x$]b000= a2 b2_ (2a2 b2) b= P1V.ON = i rectangle.
- 5.Mean Value and Probability.
Definite Integrals can be used to find the mean value of a function whose value is changing continuously by indefinitely small increments.Thus to find the mean value of f (x) for all values of x from a to b. If we divide b — a into n portions each equal to h, the mean value of the functions f (a), f (a + h ), ... f (a + n 11) is
f(a)+f(a +h)+...+f(a+n—lh)_12
h [ f(a)+f(a+h)+...+ f(a+n.—lh)]
b — a
since 'oh = b — a.If now we make it indefinitely small, we shall have the mean value of all values of f (x) from f (a) to f (b). Consequently thebmean value is b a f a f (x) dx.