Schaum's Outline of Theory and Problems of Beginning Calculus

CHAP. 33) APPLICATIONS OF INTEGRATION 11: VOLUME 265- - hetwee11 x 2 aid x 3. the volume wc want is obtained by revolving 9. about tbe y-axis Thecylindrial shell formula applies,Supplementary Problems.Sm#mt In calculating the volume of a solid of revolution we usually apply either the disk formula (or thewasher brmula) or the cylindrical shells formula (or the diffcrmcc of cylindrical shells formula). To decide whichformula to use:(I) Decide along which axis you are going to inteerate. This depends on the shape 8nd posifion of the region athat U rcvolvod.(2) (I) Use the disk formula (or the washer rormulr) if the region a is revolved pwpmdimlar to tbe axis ofintegration.(ii) Usc the cylindrical shells formula (or the difference of cytindriwl sMls formula) if the region isrevolvcd parullcl to the axis of integration.3.8 Find thc volume of the solid gmcnld-by mkng the givm-regkm aboul the @wn axis.The qion above the curve y x’. under the line y 1. and htmt’x-axis.The rcgion of par1 (a); about f he )-axis.- Thc region between the parabolas J x2x = 0 and x = 1 : about theThe region below the line y = Lr, abow the x-axis. and bctwecn x = 0 and x --I ; about rite pukand x ya ; abut ntbcr the x-axis or the paxis.The region (sec Fig. 33-17) inride the cirdc xa + y3 = r2, with 0 5 x 5 U < r; about the y-axis (Thisgiver the volume CUI from a sphere of radius r by 8 pipc d radius a whose uh b 8 diameter of thesphere.)The region (ice Fig. 33-18) inside the circk x3 + y’ = 9, with x 2 0 and y z 0. and above the liiy = a, wherc 0 $ a < r; about the y-axis. {This gives the volume of a polar cap of a sphere.)The region bounded by p - 1 + xz and y - 5; about the x-axis.Tht region (sec Fig. 33-191 inside the citde xz + (y- h)’ - U*, with 0 < a c b, about the x-axis.[Hint: When you obtain an integral d the ronn ,/- dx notice that this is the am of 8micirck of radius 0.3 This problem givts the volume of a doaghnut-shapcd soli.The region bounded by x2 = 4y and y = x/2; about the y-axis.F3g. 35.17 Fk 33.18 Fi& 33-190’I