7.2 Volume. A cross section is the shape we get when cutting straight through an object.

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1 7. Volume Let s revew the volume of smple sold, cylnder frst. Cylnder s volume=se re heght. As llustrted n Fgure (). Fgure ( nd (c) re specl cylnders. Fgure () s rght crculr cylnder. Fgure (c) s ox. A cross secton s the shpe we get when cuttng strght through n oject. For Fgure (), f we cut horzontlly (vertclly), wht s the cross secton? Answer: crcle (rectngle). For Fgure (c), f we cut horzontlly (vertclly), wht s the cross secton? Answer: rectngle (rectngle). Now, let S e the sold. Let Ax ( ) e the re of the cross secton of S n plne the x xs nd pssng through the pont x, where x. x perpendculr to Wht s the sold volume f we know the cross secton re Ax ( )? The volume of the sold s V A( x) dx The followng s the resonng. Let us consder prtton of the ntervl [, ] y ponts x x L x The plnes 0 n. x such tht x wll slce the sold nto smller sls. (lke slcng lof of

2 red.) If we choose numers x n [ x, x], we cn pproxmte the th sl S (the prt of S tht les etween the plnes x nd x ) y cylnder wth se re Ax ( ) nd heght x x x. The volume of ths cylnder s A x ( ) x, pproxmton of the volume of the th sl S, n other words, V S A x x ( ) ( ) Addng the volumes of these sls, we get n pproxmton to the totl volume: n V A x x ( ) The pproxmton ecomes etter nd etter s 0, nd the lmt s the volume of S, so n lm ( ) 0 V A x x We know the rght sde of the ove equton s the ntegrl: A ( x ) dx Therefore, we hve V A( x) dx. Smlrly, let Ay ( ) e the re of the cross secton of S n plne pssng through the pont y, where y, then the volume of S s y perpendculr to the y xs nd V A( y) dy Exmple : For sold S, the re of the cross secton perpendculr to x xs s Fnd the volume of the sold S. Soluton: A x r x x r r ( ) ( ), [, ]

3 4 V A( x) dx ( r x ) dx ( r x ) dx r r r 0 3 r r r 3 If you hve the nformton of the cross secton re, then the prolem s esy due to the formul. For the most of the questons we hve, n generl, the cross secton re s not gven drectly, ut ndrectly. In the followng, we wll look t the volume of sold of revoluton. Wht s sold of revoluton? A sold of revoluton s sold otned y rottng plne curve round some strght lne tht les on the sme plne. Fgure Wht s the sold otned f lne rotte out prllel lne? ( cylnder ) How out rght trngle rotte out round one leg? (cone) dmeter? (sphere) A semcrcle rotte out the Revew: re formuls for crcle nd nnulus. re r re R r

4 For revoluton otned y rottng y f ( x) (Fgure ), let s see wht s the re of the cross secton perpendculr to x -xs t poston x. The cross secton s crcle wth rdus s f( x ), so the re of the cross secton s So, we hve the followng: A( x) [ ( )] f x. Let S e the revoluton otned y rottng the regon ounded y y f ( x), y 0 ( x-xs), x, x out the x xs. The volume of the sold s V [ f ( x)] dx Exmple : Fnd the volume of sphere wth rdus r Remrk: We know sphere s otned y rottng semcrcle. Soluton: We let the center of the crcle s the orgn nd the rottng xs s x xs. Thus, the volume of sphere s V 4 [ f ( x )] dx [ r x ] dx r 3 r r 3 For the regon ounded y y f ( x), y g( x), x nd x ( f ( x) g( x) 0 ) rottng out the x xs. The cross secton s nnulus wth outer rdus s f( x ), nner rdus s gx ( ), so the re of the cross secton s A( x) [ f ( x)] [ g( x)]. The revoluton volume s V [ f ( x)] [ g( x)] dx

5 If the ove regon rotte out y c ( f ( x) g( x) c), then the cross secton s outer rdus s f( x) c, nner rdus s gx ( ) c, so Exmple 3: The regon ounded y the curves y volume of the resultng sold. A( x) [ f ( x) c] [ g( x) c]. The revoluton volume s V [ f ( x) c] [ g( x) c] dx x nd y x rottng out x xs. Fnd the Soluton: Frst, fnd the ntersecton ponts y solvng y x y x We hve x 0 x, y 0 y 3 V [ f ( x)] [ g( x)] dx [ x] [ x ] dx 0 0 Queston: For the regon gven Exmple 3 rottng out y, wht s the revoluton volume? (Hnt: the cross secton s nnulus wth outer rdus= x ( ) x, nner rdus = x ( ) x, so the cross secton re s A( x) [ x ] [ x ], thus Smlrly, we cn otn the volume of revoluton resultng from rottng round V [ x ] [ x ] dx 0 ). y xs. For the regon ounded y x f ( y), x g( y), y nd y ( f ( y) g( y) 0 ) rottng out the y xs. The cross secton s nnulus wth outer rdus s f( y ), nner rdus s g( y ), so the re of the cross secton s A( y) [ f ( y)] [ g( y)]. The revoluton volume s

6 V [ f ( y)] [ g( y)] dy The ove regon rottng out x c ( f ( y) g( y) c), the volume of the revoluton s V [ f ( y) c] [ g( y) c] dy Exmple 4. Determne the volume of the sold otned y rottng the regon ounded y y x nd yx out x. Solvng y x, we hve y x x x5, y0 y4 y y x x y x x y 4 Thus, the volume s 4 y 96 V [ f ( y) c] [ g( y) c] dy [ y ( )] [ ( )] dy Now, we look t the volume of some solds tht re not solds of revoluton. Exmple 5. A sold hs crculr se of rdus. rllel cross-sectons perpendculr to the se re equlterl trngles. Fnd the volume of the sold.

7 Soluton: Let s the center of the crcle (se) e the orgn, so we hve the the crcle s represented y x y. The se nd cross secton t fstnce x from the orgn re shown n the ove Fgure () nd Fgure (c). The cross secton s n equlterl trngle, so the re of cross secton s Thus, the volume of the sold s A x y y y x ( ) se heght 3 3 3( ) 3 3( ) V x dx x x 3 3 Queston. A sold hs trngulr se wth vertces (0,0), (,0) nd (0,). Cross-sectons perpendculr to the x xs re semcrcles. Fnd the volume of the sold. Soluton: Snce the two trngles re smlr, so we hve y x x y The rdus of sem crcle s y x r 4 The cross secton re, or the re of the sem crcle s A(x)= r 4 V 0 x x dx 4

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