Learning Objective 3.2A

Learning Objective 3.2A(a) Students will be able to interpret the definite integral as the limit of a Riemann sum and (b) express the limit of a Riemann sum in integral notation.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Riemann sums may be ugly, but integrals are just a cleaner looking way of writing a Riemann sum. So it's probably a good idea to learn Riemann sums.

Essential Knowledge 3.2A1 Shows you how the summation formula came to be

Essential Knowledge 3.2A2 Shows you how limits make the Riemann sum exact

Essential Knowledge 3.2A3 Shows you how standard integral notation is the same thing as a Riemann sum

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Essential Knowledge 3.2A1 Students will know that a Riemann sum, which requires the partition of an interval I, is the sum of products, each of which is the value of the function at a point in a subinterval multiplied by the length of that subinterval of the partition.

Essential Knowledge 3.2A2 Students will know that the definite integral of a continuous function f, over the interval [a, b], denoted by

, is the limit of Riemann sums as the widths of the subintervals approach 0. That is,

, where

and x_i^* is a value in the ith subinterval.

Essential Knowledge 3.2A3 Students will know that the information in a definite integral can be translated into the limit of a related Riemann sum, and the limit of a Riemann sum can be written as a definite integral.