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# Antidifferentiation as Area under a Curve

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Antidifferentiation as Area under a Curve

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.

This application is reusable. Modify the problem, then click the !!! button on the toolbar to re-execute the document to solve the new problem.

Problem Statement

Use a Riemann sum to find the area bounded by the -axis, the lines , , , and the graph of the curve corresponding to

Solution

 Enter the expression for f.

 (1)

 Use the Riemann Sums task template to solve the problem. Select Tools > Tasks > Browse > Calculus > Integration > Riemann Sums and choose one the options (Left, Right, or Midpoint).  Click Insert Default Content to insert it into the worksheet.

Calculate the Midpoint Riemann Sum

Evaluate the midpoint Riemann Sum of , .

 Enter the requisite information into the task template.  Choose the interval to be and the value of to .  Execute the rest of the task template To enter the function, use the label (1).  Enter in the interval and value of n.  Execute each of the five commands in the template.

The Midpoint Riemann Sum

Enter :

 >

 (2)

Enter the interval :

 >

 (3)

Enter the value of :

 >

 (4)

The midpoint Riemann sum:

 >

 (5)

Value of the Riemann sum:

 >

 (6)

 Find the limit of the Riemann sum as the value of goes to Use limit option in the Expression Palette to construct a limit using label (6).  Right click on the value of the limit and choose Expand.

 (7)

 (8)

Notice that if is the antiderivative of , then the area computed is , which is the content of the Fundamental Theorem of Calculus.

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.