Page 55 - Standard Handbook Of Petroleum & Natural Gas Engineering

44 Mathematics

(text continued from page 41)

Geometrically, the integral of f(x)dx over the interval [a,b] is the area bounded
by the curve y = f(x) from f(a) to f(b) and the x-axis from x = a to x = b, or the
“area under the curve from a to b.”

Properties of Definite Integrals

I: + Ip = jab

The mean value of f(x), T, between a and b is

f=-
b-a
If the upper limit b is a variable, then l:f(x)dx is a function of b and its deriva-
tive is

f(b) = -jf(x)dx db
db a
To differentiate with respect to a parameter

Some methods of integration of definite integrals are covered in “Numerical
Methods.”

Improper Integrals
If one (or both) of the limits of integration is infinite, or if the integrand
itself becomes infinite at or between the limits of integration, the integral is
an improper integral. Depending on the function, the integral may be defined,
may be equal to -=, or may be undefined for all x or for certain values of x.

Multiple Integrals

If f(x)dx = F(x)dx + C,, then