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Surface Area of a Surface of Revolution

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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 theImage buttons to watch the videos. 

Problem Statement 

Compute the surface area of the surface of revolution generated when the graph of Typesetting:-mrow(Typesetting:-mi(, is rotated about 

 

1. the Typesetting:-mrow(Typesetting:-mi(-axis 

2. the Typesetting:-mrow(Typesetting:-mi(-axis 

3. the line Typesetting:-mrow(Typesetting:-mi( 

4. the line Typesetting:-mrow(Typesetting:-mi( 

 

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  

 

Typesetting:-mrow(Typesetting:-mn(  

 

where Typesetting:-mrow(Typesetting:-mi( is a representation of the radius of revolution for a segment of the rotated curve on which Typesetting:-mrow(Typesetting:-mi( is an element of arc length. 

 

Except for the differentials Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi(, the arc length elements that appear in the solutions of these problems are 

 

 

Typesetting:-mrow(Typesetting:-mi( 

  

and  

 

   Typesetting:-mrow(Typesetting:-mi( 

 

 

Step 

Result 

Enter the expression for Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi( and press [Enter] to obtain equation labels for each. 

 

Use the definite integral template in the Expression palette to construct the expressions for Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi(. Press [Enter] after entering each expression to obtain equation labels. 

HyperlinkImage 

 

 

Typesetting:-mrow(Typesetting:-mi( 

`*`(`^`(`+`(1, `*`(4, `*`(`^`(x, 2)))), `/`(1, 2))) (3.1)
 

 

 

 Typesetting:-mrow(Typesetting:-mi( 

`+`(`*`(`/`(1, 2), `*`(`^`(`+`(4, `/`(1, `*`(y))), `/`(1, 2))))) (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 Typesetting:-mrow(Typesetting:-mi(, 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. 

 

HyperlinkImage 

Image 

 

Figure 1 The Surface of Revolution tutor applied to the surface generated by rotating Typesetting:-mrow(Typesetting:-mi(, about the Typesetting:-mrow(Typesetting:-mi(-axis. 

 

 

 

 

The surface area of the surface area of revolution is given by Typesetting:-mrow(Typesetting:-mn( where Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi(Typesetting:-mrow(Typesetting:-mo(

 

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 Typesetting:-mrow(Typesetting:-mi( ([Ctrl]+L). Press [Enter] to evaluate. Right-click on the result, Approximate>10 digits

 

HyperlinkImage 

 

 

Typesetting:-mrow(Typesetting:-mn( 

`+`(`-`(`*`(`/`(1, 4), `*`(`^`(Pi, `/`(1, 2)), `*`(`+`(`*`(`/`(1, 32), `*`(`^`(Pi, `/`(1, 2)))), `-`(`*`(`/`(1, 16), `*`(`+`(`/`(1, 2), `-`(`*`(4, `*`(ln(2))))), `*`(`^`(Pi, `/`(1, 2)))))), `-`(`*`(`/...
`+`(`-`(`*`(`/`(1, 4), `*`(`^`(Pi, `/`(1, 2)), `*`(`+`(`*`(`/`(1, 32), `*`(`^`(Pi, `/`(1, 2)))), `-`(`*`(`/`(1, 16), `*`(`+`(`/`(1, 2), `-`(`*`(4, `*`(ln(2))))), `*`(`^`(Pi, `/`(1, 2)))))), `-`(`*`(`/...
(3.1.1)
 

 

Typesetting:-mover(Typesetting:-mo( 

3.809729705 (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 Typesetting:-mrow(Typesetting:-mi(, 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.) 

 

HyperlinkImage 

 

Image 

 

Figure 2 The Surface of Revolution Tutor applied to the surface generated by rotating Typesetting:-mrow(Typesetting:-mi(about the Typesetting:-mrow(Typesetting:-mi(-axis. 

 

 

 

 

The surface area of the surface area of revolution is given by Typesetting:-mrow(Typesetting:-mn( where Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi(

Alternatively, use Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi(

 

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.  

 

HyperlinkImage 

 

 

Repeat for the alternative form of the integral. 

 

 

 

Typesetting:-mrow(Typesetting:-mn( 

`+`(`*`(2, `*`(Pi, `*`(`+`(`-`(`/`(1, 12)), `*`(`/`(5, 12), `*`(`^`(5, `/`(1, 2))))))))) (3.2.1)
 

 

Typesetting:-mover(Typesetting:-mo( 

`+`(`*`(`/`(1, 6), `*`(Pi, `*`(`+`(`-`(1), `*`(5, `*`(`^`(5, `/`(1, 2))))))))) (3.2.2)
 

 

Typesetting:-mrow(Typesetting:-mn( 

`+`(`-`(`*`(`/`(1, 8), `*`(`^`(Pi, `/`(1, 2)), `*`(`+`(`-`(`*`(`/`(20, 3), `*`(`^`(Pi, `/`(1, 2)), `*`(`^`(5, `/`(1, 2)))))), `*`(`/`(4, 3), `*`(`^`(Pi, `/`(1, 2)))))))))) (3.2.3)
 

 

Typesetting:-mover(Typesetting:-mo( 

`+`(`*`(`/`(1, 6), `*`(Pi, `*`(`+`(`-`(1), `*`(5, `*`(`^`(5, `/`(1, 2))))))))) (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 Typesetting:-mrow(Typesetting:-mi(, 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.) 

 

HyperlinkImage 

Image
Figure 3 The Surface of Revolution Tutor applied to the surface generated by rotating Typesetting:-mrow(Typesetting:-mi(about the line Typesetting:-mrow(Typesetting:-mi(

 

 

 

 

 

The surface area of the surface area of revolution is given by Typesetting:-mrow(Typesetting:-mn( where Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi(

 

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

 

HyperlinkImage 

Typesetting:-mrow(Typesetting:-mn( 

`+`(`*`(2, `*`(Pi, `*`(`+`(`-`(`/`(`*`(`/`(1, 4), `*`(`+`(`*`(`^`(Pi, `/`(1, 2))), `*`(`+`(`-`(`*`(4, `*`(ln(2)))), `-`(1)), `*`(`^`(Pi, `/`(1, 2)))), `-`(`*`(4, `*`(`^`(Pi, `/`(1, 2)), `*`(`^`(5, `/`...
`+`(`*`(2, `*`(Pi, `*`(`+`(`-`(`/`(`*`(`/`(1, 4), `*`(`+`(`*`(`^`(Pi, `/`(1, 2))), `*`(`+`(`-`(`*`(4, `*`(ln(2)))), `-`(1)), `*`(`^`(Pi, `/`(1, 2)))), `-`(`*`(4, `*`(`^`(Pi, `/`(1, 2)), `*`(`^`(5, `/`...
`+`(`*`(2, `*`(Pi, `*`(`+`(`-`(`/`(`*`(`/`(1, 4), `*`(`+`(`*`(`^`(Pi, `/`(1, 2))), `*`(`+`(`-`(`*`(4, `*`(ln(2)))), `-`(1)), `*`(`^`(Pi, `/`(1, 2)))), `-`(`*`(4, `*`(`^`(Pi, `/`(1, 2)), `*`(`^`(5, `/`...
(3.3.1)
 

 

Typesetting:-mover(Typesetting:-mo( 

14.77521436 (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 Typesetting:-mrow(Typesetting:-mi(, 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.) 

 

HyperlinkImage 

Image 

 

Figure 4 The Surface of Revolution Tutor applied to the surface generated by rotating Typesetting:-mrow(Typesetting:-mi(about the line Typesetting:-mrow(Typesetting:-mi(

 

 

 

 

The surface area of the surface area of revolution is given by Typesetting:-mrow(Typesetting:-mn( where Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi(

Alternatively, use Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi(

 

Step 

Result 

Form the integral in terms of Typesetting:-mrow(Typesetting:-mi(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

 

HyperlinkImage 

 

 

 

Repeat for the alternate form of the integral. 

 

 

 

 

Typesetting:-mrow(Typesetting:-mn( 

`+`(`*`(2, `*`(Pi, `*`(`+`(`/`(1, 12), `*`(`/`(7, 12), `*`(`^`(5, `/`(1, 2)))), `*`(`/`(1, 4), `*`(ln(`+`(9, `*`(4, `*`(`^`(5, `/`(1, 2))))))))))))) (3.4.1)
 

 

Typesetting:-mover(Typesetting:-mo(13.25453057 

 

Typesetting:-mrow(Typesetting:-mn( 

`+`(`*`(2, `*`(Pi, `*`(`+`(`-`(`/`(`*`(`/`(1, 4), `*`(`+`(`*`(`^`(Pi, `/`(1, 2))), `*`(`+`(`-`(`*`(4, `*`(ln(2)))), `-`(1)), `*`(`^`(Pi, `/`(1, 2)))), `-`(`*`(4, `*`(`^`(Pi, `/`(1, 2)), `*`(`^`(5, `/`...
`+`(`*`(2, `*`(Pi, `*`(`+`(`-`(`/`(`*`(`/`(1, 4), `*`(`+`(`*`(`^`(Pi, `/`(1, 2))), `*`(`+`(`-`(`*`(4, `*`(ln(2)))), `-`(1)), `*`(`^`(Pi, `/`(1, 2)))), `-`(`*`(4, `*`(`^`(Pi, `/`(1, 2)), `*`(`^`(5, `/`...
(3.4.2)
 

 

Typesetting:-mover(Typesetting:-mo( 

13.25453057 (3.4.3)
 

 

 

 

 

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