Properties of the Definite Integral

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Additive Interval Property
If f is integrable on the three closed intervals determined by a, b, and c, then the following is true. Notice the image below also.
  _{b                  c                  b}
§  f(x) dx = §  f(x) dx + §  f(x) dx.
  ^{a                  a                  c}


Prservation of Inequality
If f is integrable and nonnegative on the closed interval [a, b], then
         _b
0 < §  f(x) dx.
         ^a

If f and g are integrable on the losed interval [a, b] and f(x) < g(x) for ever x in [a, b], then
   _{b                  b}
§  f(x) dx < §  g(x) dx. Notice the graph below.
   ^{a                  a}

Other Properties of Definite Integrals
If f and g are integrable on [a, b] and k is a constant, then the function kf and f ± g are integrable on [a, b].
   _{b                       b}
§  kf(x) dx = k §  f(x) dx
   ^{a                       a}

   _{b                             b                  b}
§ [f(x) ± g(x)]dx = §  f(x) dx ± §  g(x) dx
   ^{a                             a                  a}