. Differential and integral calculus, an introductory course for colleges and engineering schools. y revolving the curve of exercise 32about its double tangent. 34. Find the volume generated by revolving, (1) about OX, and (2)about OY, the loop of the curve x = t2- 1, y = 5t*(t2- 1).(Art. 91, exercise 3.) 178. Solids and Surfaces of Revolution. Polar Coordinates. The necessary formulse are obtained by transforming formulae (A),(A;), (B), (B) of Arts. 175 and 176 to polar coordinates by meansof the equations x = p cos d, y = p sin 8, ds = Vp2 + p2 dd. §178 SURFACES AND SOLIDS OF REVOLUTION 263
. Differential and integral calculus, an introductory course for colleges and engineering schools. y revolving the curve of exercise 32about its double tangent. 34. Find the volume generated by revolving, (1) about OX, and (2)about OY, the loop of the curve x = t2- 1, y = 5t*(t2- 1).(Art. 91, exercise 3.) 178. Solids and Surfaces of Revolution. Polar Coordinates. The necessary formulse are obtained by transforming formulae (A),(A;), (B), (B) of Arts. 175 and 176 to polar coordinates by meansof the equations x = p cos d, y = p sin 8, ds = Vp2 + p2 dd. §178 SURFACES AND SOLIDS OF REVOLUTION 263 By this means we get y2 dx = w I p2sin2 6 d(p cos 0), S = 2tt fbyds = 2tt f psinB Vp* + p*dO, t/o t/a V = tt Tx2% = tt f p2cos20d(psin0),5 = 2tt Txds = 2tt f p cos 0 Vp2 + p2 d0. t/a t/a Example. To find the volume and surface generated by revolvingabout OX the curve p = a cos2 0. y By the foregoing formula, V = — 3 wa? L cos6 0 sin3 0 dd 2 = 3 7ra3 f (cos8 0 — cos6 d)d cos 0 IT = 3 xa3 [i cos9 0 — £ cos7 0J = —- • o -^1 Therefore the volume generated by both loops is 2 V =Further,. 4xa321 S = 2 Tra2 f a/4 - 3 cos2 0 cos3 0 d cos 0. This integral is of the form f\/4: — 3u2u3 du and can be integratedby the table of integrals. (See also exercise 14, Art. 135.) We find that J\/4- 3 v? u*du= - -^ (8 + 9w2) (4 - Su2)\135 Therefore, since u = cos 0, S = ^g [(8 + 9 cos* 0) (4 - 3 cos ,)*£- |ff (64 - 17)- ^«A 188Therefore the surface generated by both loops is 2 S = —— -kcl2. 135 264 INTEGRAL CALCULUS §179 179. Exercises. 1. Find the volume and surface generated by revolving about theinitial line the cardioid p = 2o(l — cos0). 2. Find the surface generated by revolving the lemniscate (1) about the initial line, (2) about a perpendicular to the initial linethrough the pole. 3. A loop of the curve p = a sin 2 0 is revolved about the initial lineand about a perpendicular to this line through the pole. Find the volumesgenerated. 4. Find the volume generated by revolv
Size: 2003px × 1248px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
