The Handy Math Answer Book

Thus, in the calculus terms, the area of the above graphic region is called the defi-

nite integral (also said as “the integral”) of f(x) from ato b,and is denoted by the fol-

lowing notation:

The variable xcan be replaced with any other variable. In other words, if the lim-

its of integration (aand b) are specified, it is called a definite integral, and it can be

interpreted as an area or a generalization of an area.

What are some propertiesof the definite integral?

There are several useful properties of the definite integral. Theorem oneis based on

the idea that if f(x) and g(x) are defined and continuous on [a, b], except perhaps at a

finite number of points, then the following apply:

Theorem twois based on the idea that if f(x) is defined and continuous on [a, b],

except at a finite number of points, then the following applies for any arbitrary num-

bers aand b,and any c[a, b]:

What is an indefinite integral?

From the above, we learned that when the limits of integration (in the case of aand b

above) are specified, it is called a definite integral. Contrarily, if no limits are specified,

it is called an indefinite integral. Thus, the indefinite integral is most often defined as

a function that describes an area under the function’s curve from some undefined

point to another arbitrary point. This lack of a specified first point leads to an arbitrary

constant (usually denoted as C) that is always part of the indefinite integral.

f x dx() f x dx()

b

a

a

b

##=-

f()xdx fxdx() fxdx()

a

b

a

b c

c

###=+

f()xdx 0

c

c

# =

f()xdx fxdx()

a

b

a

b

##a =a

f x() g x dx() f x dx() g x dx()

a

b

a

b

a

b

###^h+= +

fxdx()

a

b

#

230