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### Introduction

One of the topics that are studied in a first course in integral calculus is the process of approximating a given integral by various types of sums. In studying this topic, we are not just concerned with the act of finding approximations to the integral. An approximate value of an integral can be obtained by a single click on Evaluate Numerically with SWP. The purpose of this topic is to acquaint students with a variety of different kinds of sums such as left sums, right sums, trapezoidal sums, midpoint sums and Simpson sums, to point out that some of these sums will approximate a given integral more closely than others and to show that all of them provide better approximations when the interval of integration is more finely partitioned. We would like to know how much better the better sums are and how much better the sums become when the interval is more finely partitioned.

In this example, we shall show how Scientific WorkPlace can be used to study the left sums, right sums, trapezoidal sums, midpoint sums and Simpson sums of a given function f on an interval and how a course in integral calculus can thus be enriched with the help of Scientific WorkPlace.