The Handy Math Answer Book

What are antiderivativesand antidifferentiation?

An antiderivative is often interpreted as the same as an indefinite integral, but they
actually do differ in definition. Using the notations above describing the Fundamental
Theorem of the Calculus (see boxed text for definition of the Fundamental Theorem of
the Calculus), the function F(x) is an antiderivativeof f(x), described as also equal to
the integral of f(x). The actual process of finding F(x) from f(x) is called integrationor
antidifferentiation.

What is an improper integral?

An integral as seen above means that the function f(x) needs to be bounded on the
interval [a, b] (both real numbers), and the interval also must be bounded. But an
improper integral is one in which the function f(x) becomes unbounded (called a type
I improper integral) or the interval [a, b] becomes unbounded (aor b,
which is called a type II improper integral).

Are there doubleand triple integrals?

Yes, there are double and triple integrals—and even multiple integrals in equations
(, , and so on). For example, the integration of a function of three variables, w 231

MATHEMATICAL ANALYSIS

What is the Fundamental Theorem of the Calculus?

T

he Fundamental Theorem of the Calculus is the connection (or, more accu-

rately, the bridge) between the integral and the derivative; in other words, it

is another way of finding the area under a curve (see above) by evaluating the

integral. In particular, if F(x) is a function whose derivative is f(x), then the area

under the graph of yf(x) between the points aand bis equal to F(b) F(a).

This is seen as the following notations:

The number F(b) F(a) is also denoted by

Thus,

afxdx() Fx()a

b

=

b

# 6 @

6 Fx()@ab_ior F x()ab

f()xdx Fb Fa() ()

a

b

# -=