Barrons AP Calculus

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6 Definite Integrals

CONCEPTS    AND SKILLS

In  this    chapter,    we  will    review  what    definite    integrals   mean    and how to

evaluate    them.   We’ll   look    at

the all-important   Fundamental Theorem of  Calculus;

other   important   properties  of  definite    integrals,  including   the Mean

Value   Theorem for Integrals;

analytic    methods for evaluating  definite    integrals;

evaluating  definite    integrals   using   tables  and graphs;

Riemann Sums;

numerical    methods     for     approximating   definite    integrals,  including

left    and right   rectangular sums,   the midpoint    rule,   and the trapezoid

rule;

and the average value   of  a   function.

For BC  students,   we’ll   also    review  how to  work    with    integrals   based   on

parametrically  defined functions.

A. FUNDAMENTAL THEOREM OF CALCULUS (FTC);

EVALUATION OF DEFINITE INTEGRAL

If f is continuous on the closed interval [a,b] and F ′ = f, then, according to the
Fundamental Theorem of Calculus,

Definite    integral

Here is the definite integral of f from a to b; f(x) is called the integrand;

and a and b are called, respectively, the lower and upper limits of integration.