7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus


 Rafe Briggs
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1 7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. Velocity is only one type of rte in which you cn integrte to get totl. In this chpter we will discover some other uses for integrtion sed on the ide tht integrtion cuses the product of two quntities. Exmple 1: Consider the following sttement: Two steps forwrd nd one step ck. ) Wht ws the totl numer of steps tken? ) Wht is the net chnge in position? (Clled displcement) Exmple : The grph shown on the right represents the velocity of ug crwling verticlly up wll. At t = 0, the ug is 3 cm from the floor. V(t) cm/sec ) Wht is the ug s displcement from t = 0 to t = 7? ) Wht is the totl distnce the ug trvelled from t = 0 to t = 7? c) Write n integrl to represent prt nd to represent prt. t sec ) Where is the ug locted fter 7 seconds? DISPLACEMENT vs TOTAL DISTANCE Displcement: the net chnge in position; found y Totl Distnce: the totl chnge in position; found y: vt dt or vt dt. find when the oject is moving in the negtive direction, rek the integrl into pieces nd sutrct the vlue of the integrl for the re under the curve. Exmple 3: Suppose N (h) represents the rte t which tickets re written y the locl police deprtment (mesured in numer of tickets per hour). Write n integrl to find the totl numer of tickets written in dys. INTEGRAL AS A NET CHANGE: Rememer integrtion uses the process of. Which llows for wide vriety of uses. Rte = Amount of chnge from to. New Position = old position + displcement New Amount = initil mount + net chnge in mount 71
2 7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus Exmple 4: The tide removes snd from Sndy Point Bech t rte modeled y the function R given elow. A pumping sttion dds snd to the ech t rte modeled y the function S, given elow. Both R (t) nd S (t) hve units of cuic yrds per hour nd t is mesured in hours for 0 < t < 6. At time t = 0, the ech contins 500 cuic yrds of snd. 4t Rt 5sin 5 15t St 1 3t ) How much snd will the tide remove from the ech during this 6hour period? Indicte units of mesure. ) Write n expression for Y (t), the totl numer of cuic yrds of snd on the ech t time t. c) Find the rte t which the totl mount of snd on the ech is chnging t time t = 4. Explin wht this mens. d) For 0 t 6, t which time t is the mount of snd on the ech minimum? Wht is this minimum? 7. AREAS IN THE PLANE Notecrds from 7.: Are etween two curves Recll the following property of definite integrls from chpter 5: [ ] = Drw picture represented y the right hlf of this property. Explin wht the right expression mens. f ( x) g( x) f ( x) g( x) 7 
3 7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus Let s look t this region in nother wy using the pictures elow Exmple 1: Drw rectngulr strip. Wht is the height nd width of your rectngle? Would the height nd width of the rectngle strip e different if you drew it in different plce within the shded region? Is the height different for figure 1 thn it is for figure? How cn we use the rectngulr strip drwn to determine the shded re etween the two curves? Fig. 1 g Fig. g f f Are of Region Between Two Curves If f nd g re continuous on [, ] nd gx f x nd g nd the verticl lines x = nd x = is for ll x in [, ], then the re of the region ounded y the grphs of f A f x g x Exmple : Find the re of the region ounded y the grphs of y x, y x, x = 0, nd x =
4 7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus Exmple : Find the re of the region ounded y the grphs of x3 y nd x y 1. So which wy should we drw the representtive rectnglr strip? Exmple 3: Find the re of the region in the first qudrnt ounded y the grphs of y = 1 nd y 3 = x
5 7.3 Volumes Clculus 7.3 VOLUMES Notecrds from 7.3: Volume of Solid using perpendiculr cross sections, Volume of Solid using Discs, Volume of Solid using Wshers, Volume of Solid using Shells Just like in the lst section where we found the re of one ritrry rectngulr strip nd used n integrl to dd up the res of n infinite numer of infinitely thin rectngles, we re going to pply the sme concept to finding volume. The key Find the volume of ONE ritrry "slice", nd use n integrl to dd up the volumes of n infinite numer of infinitely thing "slices". Dy 1: Volumes of Solids with Known Cross Sections First Question Wht is cross section? Imgine lof of red. Now imgine the shpe of slice through the lof of red. This shpe would e cross section. Techniclly cross section of three dimensionl figure is the intersection of plne nd tht figure. It would e like cutting n oject nd then looking t the fce of where you just cut. This cross section cn e perpendiculr to the xxis or the yxis. Here's the sic ide You will e given region defined y numer of functions. We will grph tht region on n x nd y xis. Then we will ly the region flt nd uild upon tht region solid which hs the sme cross section no mtter where you slice it. Second question How do we find the volume of this solid tht hs een creted to hve similrly shped cross section, even though ech cross section my hve different size? We get to use clculus, of course! But first, we need to know how to find the Volume of one slice. Once you know the volume of one slice, you just use n integrl to dd the volumes of ll the slices to get the volume of the solid. Volume is generlized y: Are of Cross Section * thickness Exmple 1: Find the volume of the following squre "slice". Since most of the "slices" we will e deling with will hve thickness of, we will use tht sme thickness here. x Exmple : Find the volume of the following semicirculr "slice". x Volume of Solid using Perpendiculr Cross Sections: Given closed se region, uilds solid y creting slices perpendiculr to n xis. Slices of cross sectionl res re common geometric shpes. Volume found y: Are ( slice) thickness 75
6 7.3 Volumes Clculus Exmple 3: The se of solid is the region in the first qudrnt enclosed y the prol y 4x, the line x = 1, nd the x xis. Ech plne section of the solid perpendiculr to the x xis is squre. The volume of the solid is Exmple 4: The se of solid is region in the first qudrnt ounded y the xxis, the yxis, nd the line xy 8. If the cross sections of the solid perpendiculr to the x xis re semicircles, wht is the volume of the solid? Exmple 5: The se of solid is the region in the first qudrnt enclosed y the grph of y x nd the coordinte xes. If every cross section of the solid perpendiculr to the yxis is squre, the volume of the solid is given y A) y dy 0 B) y dy 0 C) 0 x D) 0 x E) 0 x 76
7 7.3 Volumes Clculus Dy : Volumes of Solids of Revolution: The Disc Method In finding the re of region, we drew n ritrry representtive rectngle. Keeping with the sme ide, if we revolve rectngle round line, it forms very thin cylinder, or disk, s shown elow. Exmple 6: Wht is the volume of the disk shown if the rdius is R nd the width of the rectngle is? Just like we did in finding the re, s we increse the numer of rectngles to infinity, the width of ech rectngle ecomes infinitely smll nd we denote this (if it is verticl strip) or dy (if it is horizontl strip). We then use n integrl to sum the volume of every one of these infinitely thin disks. This concept leds to the following: The Disc Method think circle Keys to the Disk Method: V Rx V Ry dy where R (x) nd R (y) re the rdius of the disk. Exmple 7: Drw n pproprite rectngulr strip nd find the volume of the solid formed y revolving the region out the x xis. 77
8 7.3 Volumes Clculus Exmple 8: Find the volume of the solid formed y revolving region R out the y xis. y 0 R: = x = 0 y = 16 x Exmple 9: Find the volume of the solid generted y revolving the region ounded y the grphs of the equtions xy = 6, y =, y = 6, nd x = 6 out the indicted lines. Sketch the region formed, nd drw representtive rectngulr strip for ech solid. ) the line x = 6. ) the line y =
9 7.3 Volumes Clculus Dy 3: Volumes of Solids of Revolution: The Wsher Method For the disc method, the re we rotted hd to e connected to the xis of revolution nd the representtive rectngle hd to e perpendiculr to the xis of revolution. Exmple 10: Sketch the figure formed y rotting the rectngle round the given line. Do you see why it's clled the wsher method? 5 3 Exmple 11: Wht is the volume of the figure formed ove? Generlize this volume. We will cll the outer rdius R, nd we will cll the inner rdius r. The height of the wsher formed is just the width of the strip. Just like efore, if we hve the sum of mny infinitely thin strips. The wsher method: think circle Keys to the Wsher Method: R r or where R = outer rdius nd r = inner rdius R r dy 79
10 7.3 Volumes Clculus Exmple 1: Set up nd integrl, ut do not solve, to find the volume of the solid generted y revolving the region ounded y the grphs of the equtions elow out the indicted lines. y x R: y 0 x ) the y xis ) the x xis c) the line y = 8 d) the line x = Dy 4: Volumes of Solids of Revolution: The Shell Method We hve now used two different methods to find the volume of solid formed y revolving region out line. As with the disc nd wsher methods we will egin our discussion of the shell method y considering rectngle hving width w nd length h. Wht do you notice tht is DIFFERENT out this rectngulr strip? h p : Let p e the distnce etween the xis of revolution nd the CENTER of the rectngulr strip. Exmple 13: Wht is the volume of the shell? 710
11 7.3 Volumes Clculus : If the strip is dy, then oth p nd h must e written s functions of y If the strip is, then oth p nd h must e written s functions of x dy h (y) p (y) h (x) p (x) The Shell Method Keys to the Shell Method: V pyhy dy V pxhx p is the distnce from the xis of revolution to the rectngulr strip h is the length of the strip 1 Exmple 14: Let R e the region ounded y the grphs of y, y = 0, x = 1, nd x = 4. Explin why the volume x formed y revolving R round the y xis is BEST found using the shell method insted of the disc nd wsher methods. Set up nd evlute the integrl tht gives the volume formed y revolving R round the y xis. 710
12 7.3 Volumes Clculus Exmple 15: Find the volume of the solid formed y revolving the region ounded y the xxis nd y = sin x round the yxis. Explin why it is necessry to use the shell method in this prolem. 3 Exmple 16: Find the volume of the solid formed y revolving the region ounded y the grphs of y x x 1, y = 1, nd x = 1 out the line x =. Explin why it is necessry to use the shell method in this prolem. 711
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