Math Help

Properties of Integrals (continued)

In a Nut Shell: Properties of integrals fall into two categories - standard properties (Table 1)

and comparison properties (Table 2). These properties help you evaluate integrals (standard

properties) or determine bounds on integrals based on the function or functions underneath the

integral and the limits of integration (comparison properties).

Table 2 Comparison Properties of Integrals

If f(x) ≥ 0 for a ≤ x ≤ b then ∫ f(x) dx ≥ 0

b b

If f(x) ≥ g(x) for a ≤ x ≤ b, then ∫ f(x) dx ≥ ∫ f(x) dx

a a

If m ≤ f(x) ≤ M for a ≤ x ≤ b then

m(b ˗ a) ≤ ∫ f(x) dx ≤ M (b ˗ a )

where m is the minimum value of f(x) in [a,b]

and M is the maximum value of f(x) in [a,b]