for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx


 Mark Reynold Wood
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1 Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion s n umultion proess. With few moifitions, we n exten the pplition of efinite integrls from the re of region uner urve to the re of region etween two urves. Are of Region Between Two Curves: If f n g re ontinuous on [,] n g( x) f ( x) for ll x in [,], then the re of the region oune y the grphs of f n g n the vertil lines x = n x = is [ ( ) ( )] A= f x g x x Ex: Fin the re of the region oune y the grphs of y= x +, y= x, x=0 n x=1. Are of Region Between Interseting Curves When the grphs interset the vlues for n must e ompute n re the intersetion points. Ex: Fin the re of the region oune y the grphs off( x) = x 5x 7 n g( x) = x 1
2 Ex: The sine n osine urves interset infinitely mny times, ouning regions of equl res. Fin the re of one of these regions. Curves tht interset t more thn two points If two urves interset t more thn two points, then to fin the re of the region etween the urves you must fin ll points of intersetion n hek to see whih urve is ove the other in eh intervl etermine y these points. 3 Ex: Fin the re of the region etween the grphs of f( x) = 3x x 10xng( x) = x + x If the grph of funtion of y is ounry of region, it is onvenient to use representtive retngles tht re horizontl n fin the re y integrting with respet to y. In generl, to etermine re etween two urves you n use x= [( top urve) ( ottom urve) ] x A= [( ) ( )] A= x= (vertil retngles) y= y= right urve left urve y (horizontl retngles) Ex: Fin the re of the region oune y the grphs of x= 3 y n x+ y= 1.
3 Volume: The Disk Metho n Wsher Metho Ojetive: Fin the volume of soli of revolution using the isk metho. Fin the volume of soli of revolution using the wsher metho. Fin the volume of soli with known ross setions. If region in plne is revolve out line, the resulting soli is soli of revolution, n the line is lle the xis of revolution. The simplest soli is isk, retngle revolve out n xis, n the Volume of isk = (re of the irle)(with of the isk) = πr w(where R is the rius of the isk) To relte this to other solis we n pproximte the soli using n suh isks of with x= n with rius R( x i ), n Volume of soli π n i= 1 [ ( )] R xi x This pproximtion eomes etter if we let the numer of isks n go to infinity n i i= 1 Volume of soli = lim π [ R( x)] x= π [ R( x)] x n The Disk Metho: To fin the volume of soli of revolution with the isk metho, use one of the following: Horizontl Axis of Revolution Vertil Axis of Revolution Volume = V = π [ R( x)] x Volume = V = π [ R( y)] y
4 Ex: Fin the volume of f( x) = sinxthe soli forme y revolving the region oune y the grph of the xxis. (0 x π) out the xxis. Ex: Fin the volume of the soli forme y revolving the region oune yf( x) = x n g(x) = 1 out the line y = 1. The Wsher Metho: The isk metho n e extene to over solis of revolution with holes y repling the representtive isk with representtive wsher. Volume of wsher =(volume of the lrger isk) (volume of the hole) = πrw πr w= π( R r ) w Through the sme methos s efore y ing up n wshers then letting n go to infinity, we get the Wsher Metho: Horizontl Axis of Revolution Vertil Axis of Revolution ([ R( x)] [ r( x) ]) x V = π ([ R( y [ ( ] V = π where [,] is n intervl over the xxis )] r y) ) y where [,] is n intervl over the yxis Ex: Fin the volume of the soli forme y revolving the region oune y the grphs of y= x n y= x out the xxis. Ex: Fin the volume of the soli forme y revolving the region oune y the grphs y= x + 1, y =0, x =0, n x =1 out the yxis.
5 Ex: A mnufturer rills hole through the enter of metl sphere of rius 5 in. The hole hs rius 3 in. Wht is the volume of the resulting metl ring? Volume: The Shell Metho Ojetive: Fin the volume of soli of revolution using the shell metho. Compre the uses of the isk metho n the shell metho. The vntge of the Shell Metho is we n tke retngles prllel to the xis of revolution rther thn perpeniulr to it. First we nee to look t the volume of shell. Volume of Shell = (volume of yliner) (volume of hole) w w = π( p+ ) h π( p ) h= π phw = π ( vg. rius)( height)( thikness) The Shell Metho: The volume of soli otine y rotting the region uner the grph of y = f (x) over the intervl [,] out the yxis is equl to V = π ( verge rius)( height of shell) x If we rotte the grph of x = f (y) over the intervl [,] out the xxis the volume is equl to V = π ( verge rius)( height of shell) y Where the verge rius is the istne from the xis of revolution to the enter of the representtive retngles n the height of the shell is the height of the representtive retngle. Differenes etween Disk/Wsher Methos n Shell Metho The isk/wsher methos the representtive retngles re lwys perpeniulr to the xis of revolution The shell metho the representtive retngles re lwys prllel to the xis of revolution
6 Ex: Fin the volume of the soli of revolution forme y revolving the region oune y 3 y= x x n the xxis where 0 x 1out the yxis. Ex: Fin the volume of the soli of revolution forme y revolving the region oune y the grph of x= e y n the yxis where 0 y 1out the xxis. Ex: Fin the volume of the soli of revolution forme y revolving the region oune y the grphs of y= x + 1, y= 0, x= 0, n x= 1out the yxis. Ex: Fin the volume of the soli of revolution forme y revolving the region oune y 3 the grph of y= x + x+ 1, y= 1, n x= 1out the line x =
7 Ar Length n Surfes of Revolution To fin the istne etween two points rell the formul = ( x x) ( y y) 1 1 This is iret istne etween points ut wht if we wnt to fin the istne long urve. We ll this r length. Definition of Ar Length: Let the funtion given y y = f (x) represent smooth urve on the intervl [,]. The r length of f etween n is s= 1 + [ ( )] f x x Similrly, for smooth urve given y x = g(y), the r length of g etween n is s= 1 + [ ( )] 3/ Ex: Fin the r length of y= x + 3over [0,8]. g y y Ex: Fin the r length of the prol y = x over from (0,0) to (1,1).
8 Surfe of Revolution If grph of ontinuous funtion is revolve roun line, the resulting surfe is surfe of revolution. Definition of the Are of Surfe of Revolution Let y = f (x) hve ontinuous erivtive on the intervl [,]. The re of the surfe of revolution forme y revolving the grph of f out the xxis is S π r( x) 1 [ f ( x)] x = + where r(x) is the istne etween the grph of f n the xis of revolution. Sme n e one with respet to y out vertil xis. 3 Ex: Fin the re of the surfe forme y revolving the grph of f( x) = x in the intervl [0,1] out the xxis. Ex: The urve y= 4 x, 1 x 1, is n r of the irle the surfe otine y rotting this r out the xxis. x + y = 4. Fin the re of
9 Work All physil tsk from running up hill to turning on the omputer, require n expeniture of energy. When fore is pplie to n ojet to move it, the energy expene is lle work. When onstnt fore F is pplie to move the ojet istne in the iretion of the fore, the work W is efine s fore times istne W = Fi The SI unit of fore is the newton (N), efine s 1 kgm/s. Energy n work re oth mesure in units of the joule (J), equl to 1Nm. In the British system, the unit fore is the poun, n oth energy n work re mesure in footpouns (ftl). Another unit of energy is the lorie. One ftl is pproximtely J or lories. If vrile fore is pplie to n ojet, lulus is neee to etermine the work. Work one y Vrile Fore: If n ojet is move long stright line y ontinuously vrying fore F(x), then the work W one y the fore s the ojet is move from x = to x = is = W F( x) x There re multiple methos of lulting fore. Some of the most ommon ones re Hooke s Lw (springs), Newton s Lw of Universl Grvittion (ttrtion etween prtiles), n Coulom s Lw (fore etween hrges). Hooks Lw refers to the fore F require to ompress spring n ertin istne. F = k where k is the spring onstnt whih epens on the nture of the spring. Ex: Assuming spring onstnt of k = 400 N/m, fin the work require to. Streth the spring 10 m eyon equilirium.. Compress the spring m more when it is lrey ompresse 3 m.
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