Application Center - Maplesoft
  Submit your work
Application Center Applications Area between Curves Preview

App Preview:

Area between Curves

You can switch back to the summary page by clicking here.

Learn about Maple
Download Application

 

Image 

Area Between Curves 

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 Image buttons to watch the videos. 

 

The steps in the document can be repeated to solve similar problems. 

 

Problem Statement 

Determine the value of Typesetting:-mrow(Typesetting:-mi( for which the graphs of Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( enclose an area of exactly 36. 

 

Solution 

The value of Typesetting:-mrow(Typesetting:-mi( is determined by the equation Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mo(, where Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( are the solutions of Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(. 

The difference Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( is obtained first. 

 

 

 

Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( 

`+`(a, `*`(3, `*`(x)), `-`(`*`(8, `*`(`^`(x, 2)))), `-`(3)) (3.1)
 

The limits of integration Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( are the solutions of Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(. 

Right-click the expression and select Solve > Obtain Solutions For > x 

HyperlinkImage 

 

Typesetting:-mover(Typesetting:-mo( 

`+`(`/`(3, 16), `*`(`/`(1, 16), `*`(`^`(`+`(`-`(87), `*`(32, `*`(a))), `/`(1, 2))))), `+`(`/`(3, 16), `-`(`*`(`/`(1, 16), `*`(`^`(`+`(`-`(87), `*`(32, `*`(a))), `/`(1, 2)))))) (3.2)
 

Create the definite integral, remembering to set the bounds from lowest to highest. 

From the Expression palette, click on   Typesetting:-mrow(Typesetting:-mi( . Use equation labels, or copy and paste the integrand and the limits of integration into the template fields.  Set the integral equal to 36. [Enter]. 

HyperlinkImage 

Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mo( 

`+`(`*`(`/`(1, 8), `*`(a, `*`(`^`(`+`(`-`(87), `*`(32, `*`(a))), `/`(1, 2))))), `*`(`/`(3, 2), `*`(`^`(`+`(`/`(3, 16), `*`(`/`(1, 16), `*`(`^`(`+`(`-`(87), `*`(32, `*`(a))), `/`(1, 2))))), 2))), `-`(`...
`+`(`*`(`/`(1, 8), `*`(a, `*`(`^`(`+`(`-`(87), `*`(32, `*`(a))), `/`(1, 2))))), `*`(`/`(3, 2), `*`(`^`(`+`(`/`(3, 16), `*`(`/`(1, 16), `*`(`^`(`+`(`-`(87), `*`(32, `*`(a))), `/`(1, 2))))), 2))), `-`(`...		 (3.3)
 

Simplify the equation. 

Right-click, Simplify 

HyperlinkImage 

 

`+`(`*`(`/`(1, 12), `*`(a, `*`(`^`(`+`(`-`(87), `*`(32, `*`(a))), `/`(1, 2))))), `-`(`*`(`/`(29, 128), `*`(`^`(`+`(`-`(87), `*`(32, `*`(a))), `/`(1, 2)))))) = 36 (3.4)
 

Finally, solve the equation for Typesetting:-mrow(Typesetting:-mi(. 

Right-click, Solve 

HyperlinkImage 

 

{a = `/`(663, 32)} (3.5)
 

 

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.   

 

Image