This unit introduces the concept of the integral and presents techniques
and applications
of integration. The integral is presented in two ways. First, the indefinite integral of f
is introduced as the antiderivative, which is a function F whose derivative is
f. Next,
the definite integral is introduced as a representation of the exact area under the graph of
a function. Those areas are first approximated by rectangle-based partition methods
known as Riemann sums. Then, an exact method for calculating these areas is given by
the fundamental theorem of calculus, which links together the concepts of integration and
differentiation. The unit concludes with a discussion of integration by substitution and
the second fundamental theorem of calculus.