14.3 Area Between Curves


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1 14. Ara Btwn Curvs Qustion 1: How is th ara btwn two functions calculatd? Qustion : What ar consumrs and producrs surplus? Earlir in this chaptr, w usd dfinit intgrals to find th ara undr a function and abov th x axis. By writing b a f ( x) dx or b a gxdx ( ) subtract w find th ara undr th function, abov th x axis, and btwn x a and x b. In this sction, w ll us ths aras to find th ara btwn f ( x ) and gx ( ) from x a to x b. quals W ll us this ara to dfin two important concpts in conomics, consumrs surplus and producrs surplus. 1
2 Qustion 1: How is th ara btwn two functions calculatd? In Chaptr 1, w usd dfinit intgrals to find th ara undr a function. For a positiv function f x, th dfinit intgral b a f ( x) dx corrsponds to th ara btwn f ( x ) and th x axis from x a to x b. Figur 1 Th ara undr th function f (x) and abov th x axis from x = a to x = b. Suppos w hav a smallr positiv function gx ( ) with an nclosd ara b a gxdx ( ) Figur  Th ara undr th function g (x) and abov th x axis from x = a to x = b. If w subtract ths two aras, w gt th ara of th rgion btwn th functions from x a to x b.
3 Figur Th ara btwn f (x) and g(x) from x = a to x = b. In trms of th individual aras, this rgion has an ara of b a f ( x) dx g( x) dx b Ths aras may b calculatd sparatly and thn subtractd. Howvr, it is oftn mor fficint to subtract th functions first and thn calculat th dfinit intgral with th diffrnc. a Th Ara Btwn Two Curvs Suppos th graph of f lis abov th graph of g ovr an intrval ab,. Th ara of th rgion boundd abov by f ( x ) and blow by gx ( ) from x a to x b is b a f ( x) g( x) dx Exampl 1 Find th Ara of th Enclosd Rgion Find th ara of th rgion blow. y x x
4 y x 4 Solution Th highr function is f( x) x 4 and th lowr function is gx ( ) x x. From th graph, it appars that th rgion xtnds from th point of intrsction on th lft to th point of intrsction on th right. To find th xact location of ths points, st th functions qual and solv for location of ths points, st th functions qual and solv for x: x x x x x x 4 x4 4 1 x 4, 1 Mov all trms to on sid of th quation Factor th trinomial St ach factor qual to and solv for x Ths x valus giv us th limits of intgration on th dfinit intgral. Th ara of th nclosd rgion is 4
5 f (x) g(x) 4 4 x 4 x x dx x x 4 dx 1 1 x x 4x Combin lik trms in th intgrand Tak th antidrivativ of th intgrand and substitut th limits of intgration Th ara is 15 6 or approximatly.8. Exampl Find th Ara of th Enclosd Rgion Find th ara of th rgion boundd by y x and y x. Solution Th rgion is boundd abov by f x ( ) x and blow by gx ( ) x. y x y x Th graph of th rgion appars to xtnd from x 1 to x 1. To b sur, st th functions qual and solv for x. 5
6 x x x x x 1 1 x 1, 1 Mov all trms to on sid of th quation Factor th trinomial St ach factor qual to and solv for x Now that w know th rgion xtnds from x 1 to x 1, w can us a dfinit intgral to find th nclosd ara, f (x) g(x) 1 1 x x dx x dx 1 1 x x Combin lik trms in th intgrand Tak th antidrivativ of th intgrand and substitut th limits of intgration Th ara of th rgion is 8 or approximatly In ach of th first two xampls, on function was always highr than th othr function. Whn this is th cas, th intgrand is formd by subtracting th lowr function from th highr function. If th functions cross to form th nclosd rgion, w must brak th nclosd rgion into pics. Th ara of ach of th pics is found using a dfinit intgral and th ara of ach pic is addd. Exampl Find th Ara of th Enclosd Rgion Find th ara of th rgion nclosd by y x x. and y x Solution Graph ach function to s what th nclosd rgion looks lik. 6
7 y x x y x Th graphs cross svral tims to form th nclosd rgion. St th two functions qual to find th points of intrsction: x x x x x x x x x x x x1 x, 1, Mov all trms to on sid of th quation Factor th trinomial St ach factor qual to and solv for x To find th ara of th rgion that xtnds from x 1 to x, comput th dfinit intgral highr function lowr function x x x dx x x x dx 1 1 x x x Combin lik trms in th intgrand Tak th antidrivativ of th intgrand and substitut th limits of intgration 7
8 Th ara of th portion of th nclosd rgion from x to x is x x x dx x x x dx x x x Combin lik trms in th intgrand Tak th antidrivativ of th intgrand and substitut th limits of intgration To find th ara of th ntir nclosd rgion, add th aras of th smallr rgions,, or approximatly
9 Qustion : What ar consumrs and producrs surplus? An intrsting application of th ara btwn curvs is consumrs and producrs surplus. To undrstand ths concpts, w nd to racquaint ourslvs with dmand and supply functions. W ll do this through spcific functions. A dmand function P D( Q) rlats th quantity of som product Q to th pric P that consumrs ar willing to pay for it. Suppos w hav th dmand function for milk in som rgion, DQ ( ).5Q 7.75 dollars pr gallon whr th quantity Q is in thousands of gallons. Sinc this function has a ngativ slop, as th quantity incrass th pric must dcras. For th consumr, whn th pric is chapr thy ar willing to purchas mor milk. Th supply function for milk is S Q Q dollars pr gallon This is an incrasing function sinc th producrs will supply mor lik as th pric incrass. Th dmand and supply functions intrsct at th quilibrium point Q, P P, th consumr dmands Q units and th producr is willing to supply. At a pric Q units. Th subscript rfrs to quilibrium on ach lttr. W can find th valus algbraically by stting th dmand and supply function qual. W may also stimat th valus by xamining a graph. 9
10 S Q Q DQ ( ).5Q7.75 Figur 4 Th dmand and supply function for milk in som markt. For th dmand and supply functions for milk, th markt is in quilibrium at a pric of $ pr gallon. At this pric, th consumrs ar willing to purchas thousand gallons of milk and th producrs ar willing to sll thousand gallons. This would rsult in rvnu for th producrs of dollars Rvnu thousand gallons gallon 85 thousand dollars This is th ara of th rctangl outlind by th dashd lins and axs. On this graph, ara dscribs rvnu. W could also dscrib this ara using a dfinit intgral, Q PdQ Q dq St P qual to Apply th Fundamntal Thorm of Calculus to valuat th dfinit intgral 85 This amount assums ach gallon is sold at a constant pric of $ pr gallon. 1
11 For lowr quantitis th consumrs would b willing to pay mor than $ pr gallon sinc th dmand function DQ is highr than $ dollars pr gallon. If w comput th ara undr th dmand function, w find th total amount of mony consumrs would b willing to pay for milk. Th dfinit intgral for this ara is Q DQ ( ) dq.5q7.75 dq Substitut.5Q 7.75 for DQ ( ) Q Q Apply th Fundamntal Thorm of Calculus to valuat th dfinit intgral Th consumrs would b willing to spnd thousand dollars on milk. Although consumr s would b willing to pay $51,65 for milk, thy only pay $85, by paying th quilibrium pric. Th amount savd by paying th quilibrium pric is calld th consumrs surplus Consumrs' Surplus thousand dollars This amount corrsponds to th ara btwn th dmand function DQ ( ) and th constant P from Q to Q Q. 11
12 Figur 5 Th consumrs surplus is th ara btwn th dmand function and th quilibrium pric from Q Q. Q to To hlp illustrat th maning of this ara, w hav calculat th ara blow th dmand function, blow th quilibrium pric, and subtractd th rsults. W can also find th consumrs surplus by subtracting th functions first and thn computing a dfinit intgral. Consumrs Surplus Consumrs' Surplus DQ ( ) P dq Q Exampl 4 Comput th Consumrs Surplus Us th formula abov to calculat th consumrs surplus for th milk dmand function DQ ( ).5Q 7.75 dollars pr gallon whr Q is th quantity of milk in thousands of gallons. Assum an quilibrium quantity of thousand and an quilibrium pric of $ pr gallon. 1
13 Solution Substitut th dmand function and quilibrium point into th formula, Consumrs' Surplus.5Q7.75 dq.5q4.75 dq Simplify th intgrand Q Q Apply th Fundamntal Thorm of Calculus by finding th antidrivativ Th supply function dscribs th prics at which a producr would b willing to supply a product. Th ara undr th supply function corrsponds to th amount of rvnu a producr would b willing to accpt for a product. Th ara from Q to Q is Q QdQ This mans that th producrs would b willing to accpt 14.5 thousand dollars for thousand gallons of milk. Howvr, if consumrs pay th quilibrium pric for all of this milk thy will pay a total of 85 thousand dollars. Th xcss rvnu th producrs rciv is calld th producrs surplus. Th producrs surplus, Producrs' Surplus thousand dollars corrsponds th ara btwn th constant P and th supply function SQ. 1
14 Figur 6  Th producrs; surplus is th ara btwn th quilibrium pric and th supply function from Q Q. Q to Producrs Surplus Producrs' Surplus Q P S Q dq Exampl 5 Comput th Producrs Surplus Us th formula abov to calculat th producrs surplus for th milk supply function SQ ( ) Q dollars pr gallon whr Q is th quantity of milk in thousands of gallons. Assum an quilibrium quantity of thousand and an quilibrium pric of $ pr gallon. Solution Substitut th supply function and quilibrium point into th formula, 14
15 Producrs' Surplus QdQ Q Q Apply th Fundamntal Thorm of Calculus by finding th antidrivativ 15
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