12.2 The Definite Integrals (5.2) - University of Utah

Course: Accelerated Engineering Calculus I Instructor: Michael Medvinsky

12.2 The Definite Integrals (5.2)

Def: Let f(x) be defined on interval [a,b]. Divide [a,b] into n subintervals of equal

width

x =

b-a n

,

so

x0

= a, x1 = a + x, x j

=a+

jx, xn

=b.

Let

x*j

be

an

arbitrary

(sample)

points such that x*j ( ) xj-1, xj . Then the definite integral of f from a to b is

b

n

f

(x)dx = lim f n

( )x*j x

provided

that

the

limit

exists.

If

it

douse

exist,

we

say

that

f

is

a

j=1

integrable on [a,b].

Notes:

? is an integral sign, f(x) is an integrand and a, b are lower and upper limits

of the integral respectively. Evaluating\calculating the integral is called

integration.

b

b

b

? The integral is not dependend on x, i.e. f (x)dx = f (t)dt = f (r)dr

a

a

a

? If f(x)>0 in [a,b], then an integral represent the area that lies under f(x). For

f(x) ................
................

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