Volumes of Revolution by Slicing

Transcription

1 Volumes of Revolution by Slicing Strt with n re plnr region which you cn imgine s piece of crdbord. The crdbord is ttched by one edge to stick (the xis of revolution). As you spin the stick, the re revolves nd sweeps out region in spce. xis The problem is to find the volume of revolution the volume of the region in spce which is swept out by the re. In the simplest cse the one I ve described bove you cn find the volume by cutting the region into circulr slices perpendiculr to the xis. r thickness If the rdius of slice is r, then its volume is πr 2 (thickness). In the simplest cse, the rdius is given by nonegtive function r = f(x), nd the volume is generted by revolving the re under the grph of f from x = to x = b round the x-xis. As usul, I divide the intervl x b up into pieces, with the k-th piece hving width x k. I pick n x-vlue in the k-th piece, sy x k. Then the volume of the k-th circulr slice will be The totl volume is pproximted by dding up the volumes of the slices, s you cn see in the picture bove. So if I hve n slices, then (totl volume) n k=

2 To get the exct volume, I shrink the slices, letting the thickness x k of typicl slice go to : (totl volume) = lim xk k= n The expression on the right is Riemnn sum for (totl volume) = πf(x) 2 dx. So πf(x) 2 dx. In setting up problems, I ll use shortcut rther thn writing down the Riemnn sum. I ll simply write (totl volume) = πr 2 (thickness). In the exmples I do, the xis of revolution will lwys be prllel to the x-xis or the y-xis. If the xis is prllel to the x-xis, the thickness is dx; if the xis is prllel to the y-xis, the thickness is dy. The rdius r is the distnce from the xis of revolution to the edge of the slice. It will usully be given by function specified in the problem which determines the region which is being revolved. You ll see how this works in the exmples below. Exmple. Here is the region under y = sin x from x = to x = π:.8 y sin x The region is revolved bout the x-xis. It sweeps out volume of revolution: 2

3 To find the volume, cut the solid into circulr slices like susge slices perpendiculr to the xis: The rdius of typicl slice is the height of the curve: r = sin x. The thickness is dx. Thus, the volume of typicl slice is π(sin x) 2 dx (circle re) times (thickness). So the totl volume is π π π(sin x) 2 dx = π 2 ( cos 2x)dx = π [ x ] π 2 2 sin 2x = π2 2. Exmple. The re bounded by y = 6x x 2 nd the x-xis is revolved bout the x-xis. Find the volume of the solid generted. The region is the re under the prbol y = 6x x 2 from x = to x = 6. y y = 6x - x 2 x 6 In the picture bove, I ve superimposed typicl slice over the picture of the re being revolved. You cn see tht the rdius of typicl slice is the height of the curve: r = 6x x 2. The thickness of typicl slice is dx. Thus, the totl volume is 6 π(6x x 2 ) 2 dx = π 6 [ (36x 2 2x 3 + x 4 )dx = π 2x 3 3x 4 + ] 6 5 x5 = 296π. 5 In the situtions bove, the xis of revolution ly long one edge of the re. If it does not, the volume 3

4 of revolution my hve hole in the middle: If I cut such volume into slices, I get circulr rings or wshers: r out r in The re of such wsher is πrout 2 πrin 2 = π(rout 2 rin). 2 Thus, the volume of typicl wsher is π(rout 2 r2 in ) (thickness). As before, I integrte to find the totl volume: π(r 2 out r 2 in) (thickness). Exmple. The region bounded by y = x 2 nd y = is revolved bout the x-xis. Find the volume of the solid tht is generted. y = y = x 2 - x 4

5 For typicl wsher, the inner rdius is r in = x 2 nd the outer rdius is r out =. You cn see this in the picture bove, where I ve superimposed typicl wsher on top of the picture of the region. The region extends from x = to x =. The volume is π( 2 (x 2 ) 2 )dx = π [ ( x 4 )dx = π x ] 5 x5 = 8π 5. Exmple. The region bounded by x = y 2 nd x = 2 y is revolved bout the y-xis. Find the volume generted. x = 2 - y -2 x = y 2 The curves intersect t y = 2 nd t y =. The inner rdius is r in = y 2 nd the outer rdius is r out = 2 y. The volume of the solid is 2 π ( (2 y) 2 (y 2 ) 2) [ dy = π 3 (2 y)3 ] 5 y5 = 72π 2 5. c 25 by Bruce Ikeng 5