Gols: 1. Undestnd volume s the sum of the es of n infinite nume of sufces. 2. Be le to identify: the ounded egion the efeence ectngle the sufce tht esults fom evolution of the ectngle ound n xis o foms coss section upon se the e of tht sufce. 3. Set up the definite integl fo finding the esulting volume, using e s the integnd. Homewok: Study 6.2 #1, 3, 5, 7, 11, 15, 55, 57 Fom geomety, we find volumes of edily defined geometic figues. Fo exmple: Geometic Figue Volume Sphee V = 4/3 π 3 ight Cicul Cone ight Cicul Cylinde V = 1/3 π 2 h V = π 2 h 217 1
Ou focus: Volume of cylinde ight Cicul Cylinde V = π 2 h h ight Cicul Cylinde V = π 2 h when h is smll, hve efeence ectngle h Cicle let h = Δx 217 2
evolve efeence ectngle ound the xis of evolution, pependicul to length of ectngle. Cicle let h = Δx Click gloe links evolve efeence ectngle ound the xis of evolution, pependicul to length of ectngle. Cicle let h = Δx 217 3
G: x=3, y=2, x xis, y xis F: V, x xis y=2 x=3 genetes cylinde, cn use V= π 2 h = A cicle h =π (2 2 )(3) =12π Now, use clculus: Sketch the cuves, id egion, pts of intesection Locte the xis of evolution (Aev) Hoizontl o veticl ectngle.? Aev Sketch ectngle; decide vile of integtion If veticl, use function of x nd dx. Detemine the integnd: 2 G: x=3, y=2, x xis, y xis F: V, x xis y=2 Δx =y=2 x=3 using V= π 2 h = A cicle h V =π (2 2 )(3) =12π using clculus: V = A cicle dx = π 2 dx =π 2 2 dx =4π x 3 =4π [3 ] = 12π 3 217 4
Fo nonstndd geometicl figues, no fomul, so need clculus. G: y= 4 x 2, x xis, y xis, Q I F: V, x xis Sketch the cuves, id egion, pts of intesection Locte the xis of evolution (Aev) Hoizontl o veticl ectngle.? Aev Sketch ectngle; decide vile of integtion If veticl, use function of x nd dx. Detemine the integnd: 2 geoge video Fo nonstndd geometicl figues, no fomul, so need clculus. G: y= 4 x 2, x xis, y xis, Q I F: V, x xis Δx V = A cicle dx 2 = π 2 dx = π (4 x 2 ) 2 dx = π (16 8x 2 +x 4 )dx = π[16x (8/3)x 3 +(1/5)x 5 ] = π[32 64/3+32/5 ()] = π[32+32( 2/3+1/5)] =4 x 2 = 32π[1+( 2/3+1/5)] = 256π/15 2 2 geoge video 217 5
Volumes of evolution Disk Method 1. Sketch the cuves nd identify the egion, using the points of intesection. 2. Locte the xis of evolution on the sketch. 3. Decide whethe to use hoizontl o veticl ectngle. The ectngle should e pependicul to the xis of evolution. 4. Sketch the ectngle nd detemine the vile of integtion. *If the ectngle is hoizontl, then integte with espect to y (use dy). The integnd must e in tems of y. *If the ectngle is veticl, then integte with espect to x (use dx). The integnd must e in tems of x. 5. Detemine the integnd: 2, o 2 2? *If the ectngle touches the xis of evolution, identify s the length of the ectngle. Find in tems of the ppopite vile (see ove), nd use 2 in the integnd. A = π 2 d dx A = π 2 dy c *If the ectngle does not touch the xis of evolution, identify s the distnce of the futhest end of the ectngle fom the xis of evolution nd s the distnce of the closest end of the ectngle fom the xis of evolution. Use 2 2 in the integnd. A = π ( 2 2 ) dx d A = π ( 2 2 ) dy c G: y= x, y =, x = 4 F: V, x xis Sketch the cuves, id egion, pts of intesection Locte the xis of evolution (Aev) Hoizontl o veticl ectngle.? Aev Sketch ectngle; decide vile of integtion If veticl, use function of x nd dx. Detemine the integnd: 2 o 2 2 geoge 217 6
G: y= x, y =, x = 4 F: V, y xis Sketch the cuves, id egion, pts of intesection Locte the xis of evolution (Aev) Hoizontl o veticl ectngle.? Aev Sketch ectngle; decide vile of integtion If hoizontl, use function of y nd dy. Detemine the integnd: 2 o 2 2 = d = A = π ( 2 2 ) dy c G: y= x, y =, x = 4 F: V, y xis Sketch the cuves, id egion, pts of intesection Locte the xis of evolution (Aev) Hoizontl o veticl ectngle.? Aev Sketch ectngle; decide vile of integtion If hoizontl, use function of y nd dy. Detemine the integnd: 2 o 2 2 = = d A = π ( 2 2 ) dy c d A = π ( 2 2 ) dy c 217 7
Pctice Polem Setup G: y= 2x 2, y =, x = 2 F: V, evolve out: ) y xis ) x xis c) y = 8 d) x = 2 Click on gloe ove. Then click on the "Pctice in Polem Setup" link. Volume y Coss Section Anothe ppoch to solids: 1. Stt with se: exmples include cicles, egul polygons, es defined y cuve, etc. 2. Conside polygons pependicul to the se 3. Descie the e of one of these polygons using dt fom the se 4. Integte the e coss the se to find its volume. Volume y Coss Section 217 8
G: se of solid S is ounded y y = 1 x 2 nd the x xis. Coss sections _ _ y xis e sques. F: volume of S 1. Stt with se: y = 1 x 2 2. Conside sques to se Coss Section: Sques on Pol G: se of solid S is ounded y y = 1 x 2 nd the x xis. Coss sections _ _ y xis e sques. F: volume of S x 2x 1. Stt with se: y = 1 x 2 2. Conside sques to se 3. Descie the e of the sque A = (2x) 2 = 4x 2 Coss Section: Sques on Pol 217 9
G: se of solid S is ounded y y = 1 x 2 nd the x xis. Coss sections _ _ y xis e sques. F: volume of S 1. Stt with se: y = 1 x 2 2. Conside sques to se 3. Descie the e of the sque 4. Integte the e coss the se to find its volume.??? "coss the se" mens letting y move fom y= to y=1 BUT A = f(x)! nd need A = f(y)! x y = 1 x 2 x 2 = 1 y A = (2x) 2 = 4x 2 = 4(1 y) 2x Coss Section: Sques on Pol G: se of solid S is ounded y y = 1 x 2 nd the x xis. Coss sections _ _ y xis e sques. F: volume of S 2x x Coss Section: Sques on Pol 1. Stt with se: y = 1 x 2 2. Conside sques to se 3. Descie the e of the sque 4. Integte the e coss the se to find its volume. A = (2x) 2 = 4x 2 = 4(1 y) 1 V = 4(1 y) dy 1 = 4(y y 2 ) 2 = 4(1 1/2 ) = 4/2 = 2 cuic units 217 1