Barrons AP Calculus

This important theorem says that if f is the derivative of F then the definite
integral of f gives the net change in F as x varies from a to b. It also says that we
can evaluate any definite integral for which we can find an antiderivative of a
continuous function.
By extension, a definite integral can be evaluated for any function that is
bounded and piecewise continuous. Such functions are said to be integrable.

B. PROPERTIES OF DEFINITE INTEGRALS

The following theorems about definite integrals are important.

Fundamental Theorem of  Calculus

THE FUNDAMENTAL THEOREM  OF CALCULUS:   

If  f   and g   are both    integrable  functions   of  x   on  [a,b],  then

Mean    Value   Theorem for Integrals

THE MEAN    VALUE   THEOREM FOR INTEGRALS:  If  f   is  continuous  on  [a,b]   there

exists  at  least   one number  c,  a   <   c   <   b,  such    that

By  the comparison  property,   if  f   and g   are integrable  on  [a,b]   and if  f(x)    ≤

g(x)    for all x   in  [a,b],  then

The evaluation of a definite integral is illustrated in the following examples. A
calculator will be helpful for some numerical calculations.