Volume of Solid Revolution, Arc Length, and Surface Area of Revolution By: Pragya Singh and Arielle Berman *All cartoon images and references used in this project are property of Nickelodeon. 1

. Arc Length. Surface Area. Disk. Washer. Long ago, these four calculus topics lived in harmony. Then everything changed when the AP Exam attacked. Only the Calcatar, master of all four topics, could stop it, but when the class needed him the most, he vanished. A school year has passed and my friend and I have discovered the new Calcatar, you. And although your calc - bending skills are great, you have a lot to learn before you can ace the test. But we believe you can get the 5 …. 2

Table of Contents The History of Archimedes…………………….…….4 Real World Application………………………..……..4 Volume of Solid Revolution…………………… Washer Method………………………….…….……...7 Arc Length………………………………..…………..7 Surface Area………………………………..………...7 Analytical Example: Disk Method………………..….9 Analytical Example: Washer Method………….…...10 Analytical Example: Arc Length……………..……..11 Analytical Example: Surface Area…………….…...12 AP level Multiple Choice………………….....… Conceptual Example………………………….…….15 AP level free response…………………………..…..16 Analytical Exercises……………………… AP Multiple Choice Exercises………………..… Full AP Free Response………………………..…….20 Uncle Iroh’s Wisdom…………………………..……21 Works Cited…………………………………..……..22 3

Archimedes (287 BC – 212 BC) Real World Application The measurement of volume is necessary in every aspect of life, but standard volume equations only encompass standard shapes- spheres, cubes, pyramids, etc. Volume of solid revolution allows mathematicians to find the volume of obscure objects using equations and actual measurements. This technique is extended with the “washer method”- an equation used to find the volume of a solid with a portion cut out of the center. Moreover, the arc length equation allows for the ability to find the length, and ultimately the surface area- of a complex curve and shape within certain bounds which is useful in real life to find the length and surface area of an object too big for physical measurement- such as a planet. 4 Eureka! Born in Syracuse, Sicily, then a Greek city- state, Archimedes was not only an inventor of many important devices including the water screw for raising water to irrigate the fields, but he was also an important figure in the world of mathematics throughout his life. Probably most famous for discovering mass displacement with water in a bathtub, Archimedes has had many other great accomplishments in mathematics. Although his method to find the volume of a curve revolved around an axis- originally called “Method of Exhaustion”- is now outdated, its great effects on integral calculus are still evident today.

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6 Toph may be blind and can’t read our explanations, but even she can agree that Solids of Revolution are a blast!

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8 All of this may seem a little confusing right now… Even Sokka doesn ’ t seem to understand it all. But don ’ t worry ! Ahead you will find examples of all these types of problems as well an AP level multiple choice and free response !

Analytical Example: Disk Method *Requires a Graphing Calculator Aang uses disc-like airbending! 9

The region enclosed by the graph of and the line x=4, and the x-axis is revolved about the x=0. The volume generated is The radiuses in this case would be R(x)= 04 r(x)= 0 Since the region is being revolved about the y-axis, the equation must be in terms of y. So, At x=4, y=16 since.Therefore the bounds are from 0 to 16. The integral is General form Integrate Plug in for radius Analytical Example: Washer Method 10

Analytical Example: Arc Length *Requires a Graphing Calculator Find the arc length offrom x=0 to x=4. 1.General Form of Arc length: 2.Since y=f(x), then y’= f’(x). Find the derivative of y. 3.Plug into general form. The lower and upper bounds are 0 and 4. 4.Solve. Still don’t understand? Remain calm, unlike Katara! “I’m completely calm!” 11

Analytical Example: Surface Area *Requires a Graphing Calculator Find the surface area of from x=0 to x=4 revolved about the x-axis. 1.General Form of Surface Area: 2.Since y=f(x), then y’= f’(x). Find the derivative of y. 3.Plug into general form. The left and right bounds are 0 and Now Katara starts to get it! =

AP Level Multiple Choice What is the length of the arc of from x=0 to x=3? (A) (B)7.341 (C)20.25 (D) (E) General Form Since f(x)=, then f’(x) is the derivative of y. Take the Derivative Plug into the general form. The upper bound is 3 and the lower bound is 0 You must solve the problem in order to defeat Azula. The answers are on the next slide but you better solve it right or feel her wrath! 13

14 AP Level Multiple Choice What is the length of the arc of from x=0 to x=3? (A) (B)7.341 (C)20.25 (D) (E) General Form Since f(x)=, then f’(x) is the derivative of y. Take the Derivative Plug into the general form. The upper bound is 3 and the lower bound is 0 Since this is the calculator section, you can plug the equation into your graphing calculate and you will get the answer Therefore the correct answer is D. If you forgot to square x 3 you will get If you didn’t take the derivative and plugged in for f’(x) then you will get If you forgot to use add one under the square root you get

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16 R r(x) R(x) Momo says: When attacking these problems, always try to picture the radius and how it is being revolved!

Analytical Exercises * Some May Require a Graphing Calculator 17

Exercise Section: AP Multiple Choice The Kyoshi Warriors kick butt with Calculus

Exercise Section: AP Multiple Choice (Cont.) Mai and Ty Lee are intensely studying calculus…

AP Free Response 20 k (0,1)R Get out of the way twinkle- toes!

Remember Uncle Iroh’s Wisdom 21 You are stronger and wiser and freer than you have ever been. And now you have come to the crossroads of your destiny. It’s time for you to choose. It’s time for you to choose calculus.

Works Cited Our pictures were found courtesy of: content/uploads/2009/10/aangface.jpg We also used the help of the Larson textbook and AP Central 22 Zuko says thanks you!