Surface Area of a Solid of Revolution

? 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 

Compute the surface area of the surface of revolution generated when the graph of , is rotated about 

 

1. the -axis 

2. the -axis 

3. the line  

4. the line  

 

using the Surface of Revoluation Tutor. For corroboration, compute the surface areas using first principles and compare the results to those of the Tutor. 

Solution 

In general, the surface area for a surface of revolution is given by an integral of the form  

 

 

 

where is a representation of the radius of revolution for a segment of the rotated curve on which is an element of arc length. 

 

Except for the differentials and , the arc length elements that appear in the solutions of these problems are 

 

 

 

  

and  

 

    

 

 

Step	Result
Enter the expression for and and press [Enter] to obtain equation labels for each.    Use the definite integral template in the Expression palette to construct the expressions for and . Press [Enter] after entering each expression to obtain equation labels.	(3.1)          (3.2)

 

Solution 1 

 

Step	Result
Launch and use the Surface of Revolution Tutor and compute the surface area.  Click on Tools, select Tutors> Calculus- Single Variable>Surface of Revolution. Enter , set a=0 and b=1. Select "Horizontal" for the Line of Revolution. In Plot Options, select "Constrained Scaling" and "Boxed" axes. Press [Display].  See Figure 1 below.
Figure 1 The Surface of Revolution tutor applied to the surface generated by rotating , about the -axis.

 

 

The surface area of the surface area of revolution is given by where and . 

 

Step	Result
Form the integral representing the surface area and evaluate.    Use the definite integral template in the Expression palette to construct the integral and use the equation label for ([Ctrl]+L). Press [Enter] to evaluate. Right-click on the result, Approximate>10 digits.	(3.1.1)       (3.1.2)

 

Solution 2 

 

Step	Result
Launch and use the Surface of Revolution Tutor to compute the surface area.  Click on Tools, select Tutors> Calculus- Single Variable>Surface of Revolution. Enter , set a=0 and b=1. Select "Vertical" for the Line of Revolution. In Plot Options, select "Constrained Scaling" and "Boxed" axes. Press [Display].  (See Figure 2 below.)
Figure 2 The Surface of Revolution Tutor applied to the surface generated by rotating about the -axis.

 

 

The surface area of the surface area of revolution is given by where and . 

Alternatively, use and . 

 

Step	Result
Form the integral and evaluate.     Use the definite integral template from the Expression palette to construct the integral. Remember to change the variable of integration from x to y. Press [Enter]. Right-click, Simplify.           Repeat for the alternative form of the integral.	(3.2.1)       (3.2.2)       (3.2.3)       (3.2.4)

 

Solution 3 

 

Step	Result
Launch and use the Surface of Revolution Tutor to compute the surface area.  Click on Tools, select Tutors> Calculus- Single Variable>Surface of Revolution. Enter , set a=0 and b=1. Select "Horizontal" for the Line of Revolution and set the distance of rotation line to axis to 2. In Plot Options, select "Constrainted Scaling" and "Boxed" axes. Press [Display].  (See Figure 3 below.)
Figure 3 The Surface of Revolution Tutor applied to the surface generated by rotating about the line .

 

 

The surface area of the surface area of revolution is given by where and . 

 

Step	Result
Form the integral and evaluate.     Use the definite integral template from the Expression palette to construct the integral. Press [Enter] to evaluate the integral. Approximate the result: right-click on the result, Approximate>10 digits.	(3.3.1)       (3.3.2)

 

Solution 4 

 

Step	Result
Launch and use the Surface of Revolution Tutor to compute the surface area.  Click on Tools, select Tutors> Calculus- Single Variable>Surface of Revolution. Enter , set a=0 and b=1. Select "Vertical" for the Line of Revolution and set the distance of rotation line to axis to 2. In Plot options, select "Boxed" axes. Press [Display].  (See Figure 4 below.)
Figure 4 The Surface of Revolution Tutor applied to the surface generated by rotating about the line .

 

 

The surface area of the surface area of revolution is given by where and . 

Alternatively, use and . 

 

Step	Result
Form the integral in terms of and evaluate.     Use the definite integral template from the Expression palette to construct the integral. Press [Enter]. Approximate the result.Right-click on the result, Approximate>10 digits.            Repeat for the alternate form of the integral.	(3.4.1)           (3.4.2)       (3.4.3)

 

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