Mathematical Formulas for Industrial and Mechanical Engineering

G

If f(x) is a continuous function on (2N, b], then

ð

b

2N

f ðxÞdx 5 lim

n!2N

ð

b

n

f ðxÞdx

Note: The previous improper integrals are convergent if the limits exist and are finite;

otherwise the integrals are divergent.

G

ð

N

2N

f ðxÞdx 5

ð

c

2N

f ðxÞdx 1

ð

N

c

f ðxÞdx

If for some real number c, both of the integrals in the right side are convergent, then

the integral

ð

N

2N

f ðxÞdx is also convergent; otherwise it is divergent.

G

Comparison theorems: Let f(x)andg(x) be continuous functions on the closed interval [a, N).

Suppose that 0 # g(x) # f(x)forallx in [a, N).

5.16 Continuity of a Function

The concept of a continuous function is that it is a function, whose graph has