Riemann Sums
Copyright Maplesoft, a division of Waterloo Maple Inc., 2007
Introduction
This application is one of a collection of examples teaching Calculus with Maple. These applications use Clickable Calculus� methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips. Click on the buttons to watch the videos.
Problem Statement
Rieman sums approximate area under a curve by accumulating the areas of rectangles. They are the basis for the definition of the Riemann (definite) integral, and can be used to approximate the value of a definite integral.
Use a Riemann sum to find the area bounded by the axis and the graph of the curve corresponding to .
First, load the Student Calculus 1 package.

The package can be found at Tools>Load Package> Student Calculus 1.


Loading
Student:Calculus1
Enter the expression for :

Type the expression then press [Enter].



(2.1) 
Solution
Obtain the intercepts.

Enter the equation label for f(x) ([Ctrl][L] ) on a new line and press Enter. Then right click on the result and choose Solve > Solve.



(3.1) 

(3.2) 
Graph the expression.

Enter the equation label for f(x) on a new line and press Enter. Then right click on the result and choose Plots > Plot Builder. Select the x range as the intercepts found above.



(3.3) 
Launch the Riemann Sum tutor and use the intercepts for and .

Again, enter the equation label for f(x) and press [Enter]. Next, rightclick on the result and choose Tutors > Calculus  Single Variable > Riemann Sums and set the range from 2 to 3.



(3.4) 
The Tutor provides the approximate area under the curve. Six different types of approximating rectangles are available, each distinguished by the location of the point at which is evaluated. Clicking the Animate button triggers an animation showing the effect of increasing the number of rectangles.
Related Resource
A Task template for the Riemann Sum can be found in the following location:
Tools>Tasks>Browse>Calculus>Integration>Riemann Sums and choose one of the options (Left, Right, or Midpoint).
Legal Notice: The copyright for this application is owned by Maplesoft. The application is intended to demonstrate the use of Maple to solve a particular problem. It has been made available for product evaluation purposes only and may not be used in any other context without the express permission of Maplesoft.