LEARNING FROM MISTAKES
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1 AP Central Quetion of te Mont May 3 Quetion of te Mont By Lin McMullin LEARNING FROM MISTAKES Ti i te firt Quetion of te Mont tat ill appear on te Calculu ection of AP Central. Tee are not AP Exam quetion, but rater quetion on topic related to beginning calculu tat, I ope, you ill find intereting and intructive, and tat you can are it your tudent to give tem jut a little deeper undertanding of matematic. Since finding volume by te metod of Cylindrical Sell i important, but no longer on te AP Coure Decription, teacer often preent it during te aftermat te day after te AP Exam are given. So our firt Quetion of te Mont i about Cylindrical Sell. It i baed on quetion 5 on te 1996 AB exam te function a canged, but te idea remain te ame. Te exam quetion a a volume problem (ic a intended to be done by te aer metod). Ti a folloed by an Accumulation Function / Related Rate quetion tat ued te reult of te previou part. Recently a teacer gave te original quetion and one of er tudent found te volume uing Cylindrical Sell, no problem tere. But ten, uing Sell, te related rate computation came out very rong. Te key to undertanding te mitake i undertanding at you are actually finding en you ork it Cylindrical Sell. Here our ituation and your problem for ti mont: A ater tank a te ape on belo, obtained by rotating te curve y = x from x = to x = 3 around te y-axi, ere x and y are meaured in feet. Water flo into te tank at te rate of 5 cubic feet per minute. y x 3 Lin McMullin, All Rigt Reerved Page 1 of 5
2 (A) Find te volume of ater in te tank uing te Waer metod. Find te volume of ater in te tank uing te metod of Cylindrical Sell. Indicate unit of meaure for bot. Compare your aner. (Tey ould be te ame of coure.) (B) Let be te dept of ater in te tank. Uing a definite integral, rite to function tat give te volume of ater in te tank a a function of, one, V ( ), uing te Waer metod and te oter, V ( ), uing te metod of Cylindrical Sell. (C) Ho fat i te dept of ater in te tank increaing en =? Indicate unit of meaure. Again do ti part uing bot te Waer and Cylindrical Sell metod. Compare your aner. If tey are te ame, congratulation! You may take te ret of te day off. If tey re not te ame, find out y not. (Hint: Look at te picture you ued for te Sell metod in (b); o i te ater accumulating?) Do all tree part of te quetion before cecking te aner and explanation! aner and explanation = Link Aner and Explanation. End Quetion 3 Lin McMullin, All Rigt Reerved Page of 5
3 Aner and Explanation: 9 9 cubic feet. (a) Waer: V = x dy = y dy = 4.5 Sell: 3 3 (9 ) (9 ) 4.5 cubic feet. V = x y dx = x x dx = (b) Waer: V ( ) = x dy = y dy Here te intereting part: For te volume by ell, ( ) ( 9 ) V x x dx. Te temptation i to do te Sell part te ame ay a te Waer part. Ti doen t ork becaue of te ay ell are et up. Ti integral give te volume of ater currently in te tank plu te ater in a cylinder itting on top of te ater in te tank. (Te cylinder a a radiu of and a eigt of 9.) Ti i a if te ater ere filling an expanding cylinder running from te top of te tank don to te current ater level. Water doe not beave tat ay en you pour it into a tank. Wen te tank i completely full te volume i te ame, but it i not te ame ile te tank i filling. [Caption:] Te partially filled tank? Tere are to correct function uing te Cylindrical Sell approac for te accumulating volume. Firt, te variable eigt o up in te integrand a ell a te limit of integration: 3 Lin McMullin, All Rigt Reerved Page 3 of 5
4 1 ( ) = ( ) = ( ). V x y dx x x dx Or te extra cylinder above te current level of ater can be ubtracted: ( ) = ( 9 ) ( ) ( 9 ) V x x dx. (c) By Waer uing te Fundamental Teorem of Calculu: Recall tat V ( ) = y dy, o terefore dv d =, and en = and dv d 5 = 5, = feet per minute. By Sell: Firt, if you reaoned ( ) = ( 9 ) V x x dx dv ( ( ) ) 1 d = 9 reread Here te intereting part above again. Oterie, uing te Fundamental Teorem of Calculu: dv d = x 9 x dx 9 ( ) ( ) ( ) ( ) 1 1/ ( ) ( ) = 9 9 d = [ ] d = d Ten, en = and dv d 5 Te function 1 = ( ) = 5, = feet per minute. cannot be differentiated directly uing te Fundamental V x x dx Teorem of Calculu becaue te independent variable appear in te integrand. More advanced metod are required. Finally, te eay ay to do te problem i to evaluate any one of te integral in (b): dv d V ( ) = V1 ( ) = V ( ) =. Ten = and te ret i eay. 3 Lin McMullin, All Rigt Reerved Page 4 of 5
5 Doug Kulmann, of Pillip Academy Andover, a te firt to give a cogent explanation of te original mitake en te quetion a firt aked on te AP Calculu Electronic Dicuion Group. Tank you Doug! 3 Lin McMullin, All Rigt Reerved Page 5 of 5
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