Riemann integrals are introduced as a concept using the example of finding the area of a circle from the areas of N-sided polygons inscribed in the circle. Signed area (positive above the x-axis, negative below) is introduced.

Interval partitions are defined, including the concepts of mesh size and fine vs. coarse partitions.

Definition, including a discussion of partition choices when computing these sums.

Definite integrals are defined. Includes an example using the function f(x) = x.

Using Riemann sums to find the Riemann integral for the function f(x) = e^x.

Using Riemann sums to find the Riemann integral for the function f(x) = x^r.

Definition of the definite integral as the area under a curve, including definition of the integrand.

Definition, including the use of Riemann sums in finding the area under a curve.

Using upper sums to evaluate a definite integral.

Solution (PDF) Page 3

Three problems which involve evaluating Riemann sums and integrals.

Solution (PDF)

Evaluating an integral using the definition of an integral as the limit of sums.

Solution (PDF)^# Page 1

Seven questions which involve using sigma notation for sums, computing Riemann sums for definite integrals, and evaluating limits by relating them to Riemann sums.

Solution (PDF - 4.0 MB) Page 39