# [ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves

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1 Gols: 1. To find the re etween two curves Section 6.1 Are of Regions etween two Curves I. Are of Region Between Two Curves A. Grphicl Represention = _ B. Integrl Represention [ ( ) ( )] f x g x dx = C. Exmple 1. Find the re of the region ounded the grphs of x = 0, nd x = 1. = x +, x =,. Find the re etween = cos x nd = sin x from 0 to π. II. Are of region etween intersecting curves A. Exmples Find the re of the region etween the grphs of f ( x) x x = +. f ( x) 3x x 10x = nd

2 B. Horizontl vs Verticl Representtion 1. Exmple: ) Find the re of the region ounded = ± 3 x nd = x 1 using verticl representtion. 4 - ) Find the sme re ove ut using horizontl representtion. (1) Convert to x = x = 3 nd x = + 1 () Integrte in reltion to. Find the re etween = ± x, = x Homework: p ll, 14, 19, 7, odds, 41, 47, 49, 71, 79, 80

3 Section 6. Volume the Disk Method Gols: 1. To find the volume of solid of revolution using the disk method.. To find the volume of solid of revolution using the wsher method. 3. To find the volume of solid of revolution using known cross section. I. The Disk Method A. Grphicl Representtion of function rotted round the x-xis. 1. Volume of one disk of rottion out the x-xis r =. Volume of mn disks rotted out the x-xis h = x 3. Exmples ) Find the volume of region ound = x, 0 x 4, nd the x- xis rotted out the x-xis. ) Find the volume of region enclosed the semicircle nd the x-xis rotted out the x-xis. x + = 4 c) Find the volume generted revolving the region ounded = x nd the lines = 1 nd x = 4 out the line = 1.

4 B. Revolving out the -xis Volume of disk rotted out the -xis - r = x h =. Volume of mn disks rotted out the -xis 3. Exmple: 1 Find the volume of the region ounded =, 1 4, nd the -xis x rotted out the -xis. II. The Wsher Method A. Grphicl Representtion 4 - B. Volume of single wsher C. Volume of mn wshers R = 1 r = h = x

5 D. Exmples 1. Find the volume of the region ounded out the x-xis. = x + 1 nd x 3 = + revolved. Find the volume of the region ounded out the line = 5 = x + 1 nd x 3 = + revolved 3. Find the volume of the region ounded the -xis. = x nd = x revolved out 4. Find the volume of the region ounded the line x =. = x nd = x revolved out

6 III. Find volume of Solid with Known Cross Sections A. Grphicl Representtion B. For cross-sections of re A( x) tken perpendiculr to the x-xis, Exmple: Find the volume of solid whose se is the region ounded the lines x x = 1, = 1+, nd x = 0, with cross sections tken perpendiculr to the x-xis re squres. C. For cross sections of re A( ) tken perpendiculr to the -xis, Exmple: Find the volume of solid whose se is ound the grph = 4 x, -xis, nd the x-xis, with cross sections tken perpendiculr to the -xis re semicircles. Homework: D 1: p ll, 7-10 ll, 3, 7, ll, 51 D : p. 48 5, 6, 11, 13, 0, 1, 55, 57, D 3: p , 60

7 Section 6.3 Volume the Shell Method Gols: 1. To fond the volume of solid of revolution using the shell method. I. The Shell Method A. Grphicl Representtion of function rotted out the x-xis. 1. Volume of one shell = volume of clinder volume of hole See exmple: x = f ( ) Axis of Rottion x Axis of Rottion = f ( x). Volume of mn shells 3. Exmple: ) Find the volume of the solid of revolution formed revolving the region ounded = x,the x-xis, nd x = 4 out the -xis x - -4

8 II. Comprison of Disc nd Shell A. Summr see pge 49 B. Exmple: Find the volume of the solid formed revolving the region ounded the grphs of = x + 1, = 0, x = 0, nd x = 1 1. Find using disk method x. Find using shell method Homework: p ll, 7, 13, 17, 1, 30, 4, 43

9 Section 6.4 Arc Length nd Surfce of Revolution Gols: 1. To compute the length of curve.. To find the surfce re of revolution out the x- or -xis. c I. Arc Length A. Length of one secnt line: x B. Sum of the lengths of severl secnt lines: C. Exmple: 1. A portion of roller coster trck is modeled the grph of = sin(0.03 x) + 40 cos(0.0x 1) from 0 x 68 feet. Find the length of the trck. x. Find the length of the curve = from 0 x. Note discontinuit of t x = 0. Wht cn we do? 3

10 II. Surfce Are A. Using one ring: S = C L or S = π rl L is rc length L r = Axis of Rottion B. Sum of mn rings 1. Rotted out the x-xis:. Rotted out the -xis: C. Exmples 1. Find the re of the surfce generted revolving the curve = x, 1 x out the x-xis.. 3. Find the surfce re generted revolving line segment = 1 x, 0 1 out the -xis. Homework: p ,7,11,15,18,7,33,35, ll,49

11 Section 6.5 Work Gols: 1. To compute the work done in vriet of rel world pplictions. I. Work when CONSTANT force is pplied A. Definition: If n oject is moved distnce d in the direction of n pplied constnt force then, W = Fd. B. Exmple: Determine the work done lifting 50-l oject 4 feet. II. Work when VARIABLE force is pplied A. Definition: If n ject moved long stright line continuousl vring force =. F( x ) from point to point then, W F( x) dx B. Exmples 1. A force of 750 pounds compresses spring 3 inches from its nturl length of 15 inches. Find the work done in compressing the spring n dditionl 3 inches. Note: Hooke s Lw: F = kx x units form nturl length. K constnt for tht prticulr spring.. How much work does it tke to pump the wter from n upright nd full right circulr clindricl tnk of rdius 5-ft nd height 10-ft to level of 4-ft ove the top of the tnk? Note: F = wv w weight of 1 cuic foot of liquid wter weight is 6.5-ls V volume Wter level Homework: p ll, 7, 9, 11, 17, 19, 1, 5, 35, 37

12 Section 6.7 Fluid Pressure nd Fluid Force Gols: 1. To compute the force exerted on sumerged surfce fluid. I. The Constnt-Depth Formul for Fluid Force A. Definition the force per unit of re over the surfce of od. B. Formul: P = wh w weight-densit of the liquid per unit of volume h depth of liquid C. Pscl s Principle pressure exerted fluid t depth h is trnsmitted equll in ll directions (I.e., pressure t the indicted depth is the sme for ll three ojects.) D. The fluid force on sumerged horizontl surfce of re A is: F = PA = wha E. Exmple: Find the fluid force on rectngulr metl sheet mesuring 3-ft 4-ft tht is sumerged in 6 feet of wter. II. The Vrile-Depth Formul for Fluid Force A. Formul: F = w f ( ) d w weight-densit of the liquid per unit of volume depth of liquid f ( ) horizontl length d thickness B. Exmple: 1. Find the fluid force on the verticl side of tnk, where the se is 8 feet nd the height is 6 feet. Assume the tnk is full of wter.

13 . A verticl gte in dm hs the shpe of rectngle 8-ft cross the top with height of 5-ft. Wht is the fluid force on the gte if the top of the gte is 4-ft elow the surfce of the wter? 3. A verticl gte in dm hs the shpe of n isosceles trpezoid 8-ft cross the top nd 6-feet cross the ottom, with height of 5-ft. Wht is the fluid force on the gte if the top of the gte is 4-ft elow the surfce of the wter? 4. A circulr oservtion window on mrine science ship hs rdius of 1-ft, nd the center of the window is 8-ft elow wter level. Wht is the fluid force on the window? Homework: p nd ll

14 Section 6.8 Other Modeling Applictions Gols: 1. To compute the totl distnce trveled prticle long coordinte plne.. To compute the position of prticle long line. Totl Distnce Trveled: v ( t ) dt Position: v ( t ) dt Exmple: A prticle moves long the x-xis so tht the velocit t time t, 0 t 5, is given the v( t) = 3( t 1)( t 3). At t = the position of the prticle is s () = 0. Find the minimum ccelertion. Find the totl distnce trveled. Find the position t t = 5. Find the verge velocit over the intervl x 4 6

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