用二重積分計(jì)算旋轉(zhuǎn)體體積的幾何解釋

1、 July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 1 July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 2 設(shè)設(shè)D是上半平面內(nèi)的一個(gè)有界閉區(qū)域。是上半平面內(nèi)的一個(gè)有界閉區(qū)域。 將將D繞繞x軸軸旋轉(zhuǎn)一周得一旋轉(zhuǎn)體，求該旋旋轉(zhuǎn)一周得一旋轉(zhuǎn)體，求該旋轉(zhuǎn)體的體積轉(zhuǎn)體的體積V。 我們用元素法來(lái)建立旋轉(zhuǎn)體體積的二我們用元素法來(lái)建立旋轉(zhuǎn)體體積的二重積分公式。重積分公式。D July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 3d( , )x

2、yD在區(qū)域在區(qū)域D的的(x,y)處取一個(gè)面積元素處取一個(gè)面積元素d它到它到x軸的距離是軸的距離是 y （如圖）。（如圖）。該面積元素繞該面積元素繞x軸軸旋轉(zhuǎn)而成的旋轉(zhuǎn)體的體積約為：旋轉(zhuǎn)而成的旋轉(zhuǎn)體的體積約為：2dVyd（體積元素）（體積元素）于是整個(gè)區(qū)域繞于是整個(gè)區(qū)域繞x軸軸旋轉(zhuǎn)而旋轉(zhuǎn)而成的旋轉(zhuǎn)體的體積為：成的旋轉(zhuǎn)體的體積為：2DDVydVdy July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 4d( , )x yD命題命題1：上半平面內(nèi)一個(gè)有界閉區(qū)域：上半平面內(nèi)一個(gè)有界閉區(qū)域D繞繞x軸軸旋轉(zhuǎn)而成的旋轉(zhuǎn)體的體積為：旋轉(zhuǎn)而成的旋轉(zhuǎn)體的體積為

3、：2DdVyy July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 5下面來(lái)解釋以上公式的幾何意義下面來(lái)解釋以上公式的幾何意義 July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 6區(qū)域區(qū)域D中一面積元素中一面積元素 繞繞x軸軸旋轉(zhuǎn)而成旋轉(zhuǎn)而成的旋轉(zhuǎn)體為一的旋轉(zhuǎn)體為一環(huán)形體環(huán)形體(如圖如圖)。dd July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 7區(qū)域區(qū)域D中一面積元素中一面積元素 繞繞x軸軸旋轉(zhuǎn)而成旋轉(zhuǎn)而成的旋轉(zhuǎn)體為一環(huán)形體的旋

4、轉(zhuǎn)體為一環(huán)形體(如圖如圖)。d其體積約為：其體積約為：2dVyd（體積元素）（體積元素） July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 8將將dV在在D上二重積分的幾何意義是將劃分上二重積分的幾何意義是將劃分D的的n個(gè)面積元素分別繞個(gè)面積元素分別繞x軸旋轉(zhuǎn)而成的旋轉(zhuǎn)體軸旋轉(zhuǎn)而成的旋轉(zhuǎn)體相加，得到整個(gè)相加，得到整個(gè)D繞繞x軸旋轉(zhuǎn)的旋轉(zhuǎn)體。軸旋轉(zhuǎn)的旋轉(zhuǎn)體。于是整個(gè)區(qū)域繞于是整個(gè)區(qū)域繞x軸軸旋轉(zhuǎn)而成的旋轉(zhuǎn)體旋轉(zhuǎn)而成的旋轉(zhuǎn)體的體積為：的體積為：2DDVydVd July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體

5、體積計(jì)算公式的幾何意義 9以下圖形給出了這種方法的幾何解釋以下圖形給出了這種方法的幾何解釋 July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 10 July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 11display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4

6、_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h_20,h_28,h_2_8,h_44,h_4_4,h_66,h_6_6,h_80,h_84,h_8_4,scaling=constraine

7、d,color=green); July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 12 display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28

8、,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h_20,h_28,h_2_8,h_44,h_4_4,h_66,h_6_6,h_80,h_84,h_8_4,scaling=constrained,color=green); July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 13 display(xzhou

9、,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y

10、_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h_20,h_28,h_2_8,h_44,h_4_4,h_66,h_6_6,h_80,h_84,h_8_4,scaling=constrained,color=green); July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 14 July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 15設(shè)想用電纜做成一個(gè)圓環(huán)體設(shè)想用電纜做成一個(gè)圓環(huán)體那么這個(gè)圓環(huán)體可由電纜中很多圓環(huán)形那么

11、這個(gè)圓環(huán)體可由電纜中很多圓環(huán)形狀的光纖組成狀的光纖組成因此，我們可以把這種計(jì)算旋轉(zhuǎn)體體積因此，我們可以把這種計(jì)算旋轉(zhuǎn)體體積的方法形象地稱(chēng)為的方法形象地稱(chēng)為光纖法光纖法或或電纜法電纜法 July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 16更多的圖形更多的圖形 July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 17 July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 18 July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)

12、算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 19with(plots):xzhou:=spacecurve(x,0,0, x=-2.2, thickness=1,color=black):yzhou:=spacecurve(0,y,0, y=-2.2, thickness=1,color=black):zzhou:=spacecurve(0,0,z, z=-2.4, thickness=1,color=black):a:=0:b:=3:R:=1:r:=0.1:yuan:=spacecurve(0,a+R*cos(t),b+R*sin(t), t=0.2*Pi, thickness=3,col

13、or=red):a0:=0:a2:=0.2:a4:=0.4:a6:=0.6:a8:=0.8:a_2:=-0.2:a_4:=-0.4:a_6:=-0.6:a_8:=-0.8:b0:=3:b2:=3.2:b4:=3.4:b6:=3.6:b8:=3.8:b_2:=3-0.2:b_4:=3-0.4:b_6:=3-0.6:b_8:=3-0.8:y00:=spacecurve(0,a0+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y02:=spacecurve(0,a0+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y04:

14、=spacecurve(0,a0+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y06:=spacecurve(0,a0+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y08:=spacecurve(0,a0+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y0_2:=spacecurve(0,a0+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y0_4:=spacecurve(0,a0+r*cos(t),b_4+r*sin(t), t

15、=0.2*Pi,color=blue):y0_6:=spacecurve(0,a0+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y0_8:=spacecurve(0,a0+r*cos(t),b_8+r*sin(t), t=0.2*Pi,color=blue):y20:=spacecurve(0,a2+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y22:=spacecurve(0,a2+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y24:=spacecurve(0,a2

16、+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y26:=spacecurve(0,a2+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y28:=spacecurve(0,a2+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y2_2:=spacecurve(0,a2+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y2_4:=spacecurve(0,a2+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=bl

17、ue):y2_6:=spacecurve(0,a2+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y2_8:=spacecurve(0,a2+r*cos(t),b_8+r*sin(t), t=0.2*Pi,color=blue):y40:=spacecurve(0,a4+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y42:=spacecurve(0,a4+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y44:=spacecurve(0,a4+r*cos(t),b4+r*s

18、in(t), t=0.2*Pi,color=blue):y46:=spacecurve(0,a4+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y48:=spacecurve(0,a4+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y4_2:=spacecurve(0,a4+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y4_4:=spacecurve(0,a4+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y4_6:=spacec

19、urve(0,a4+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y4_8:=spacecurve(0,a4+r*cos(t),b_8+r*sin(t), t=0.2*Pi,color=blue):y60:=spacecurve(0,a6+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y62:=spacecurve(0,a6+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y64:=spacecurve(0,a6+r*cos(t),b4+r*sin(t), t=0.2*Pi,

20、color=blue):y66:=spacecurve(0,a6+r*cos(t),b6+r*sin(t), t=0.2*Pi,color=blue):y6_2:=spacecurve(0,a6+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y6_4:=spacecurve(0,a6+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y6_6:=spacecurve(0,a6+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y80:=spacecurve(0,a8+r*cos

21、(t),b0+r*sin(t), t=0.2*Pi,color=blue):y82:=spacecurve(0,a8+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y84:=spacecurve(0,a8+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y8_2:=spacecurve(0,a8+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y8_4:=spacecurve(0,a8+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y8

22、_6:=spacecurve(0,a8+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y_20:=spacecurve(0,a_2+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y_22:=spacecurve(0,a_2+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y_24:=spacecurve(0,a_2+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y_26:=spacecurve(0,a_2+r*cos(t),b6+r*s

23、in(t), t=0.2*Pi,color=blue):y_28:=spacecurve(0,a_2+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y_2_2:=spacecurve(0,a_2+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y_2_4:=spacecurve(0,a_2+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y_2_6:=spacecurve(0,a_2+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y_

24、2_8:=spacecurve(0,a_2+r*cos(t),b_8+r*sin(t), t=0.2*Pi,color=blue):y_40:=spacecurve(0,a_4+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y_42:=spacecurve(0,a_4+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y_44:=spacecurve(0,a_4+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y_46:=spacecurve(0,a_4+r*cos(t),b6+r

25、*sin(t), t=0.2*Pi,color=blue):y_48:=spacecurve(0,a_4+r*cos(t),b8+r*sin(t), t=0.2*Pi,color=blue):y_4_2:=spacecurve(0,a_4+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y_4_4:=spacecurve(0,a_4+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y_4_6:=spacecurve(0,a_4+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):

26、y_4_8:=spacecurve(0,a_4+r*cos(t),b_8+r*sin(t), t=0.2*Pi,color=blue):y_60:=spacecurve(0,a_6+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y_62:=spacecurve(0,a_6+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y_64:=spacecurve(0,a_6+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y_66:=spacecurve(0,a_6+r*cos(t),b6

27、+r*sin(t), t=0.2*Pi,color=blue):y_6_2:=spacecurve(0,a_6+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y_6_4:=spacecurve(0,a_6+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y_6_6:=spacecurve(0,a_6+r*cos(t),b_6+r*sin(t), t=0.2*Pi,color=blue):y_80:=spacecurve(0,a_8+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue

28、):y_82:=spacecurve(0,a_8+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y_84:=spacecurve(0,a_8+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y_8_2:=spacecurve(0,a_8+r*cos(t),b_2+r*sin(t), t=0.2*Pi,color=blue):y_8_4:=spacecurve(0,a_8+r*cos(t),b_4+r*sin(t), t=0.2*Pi,color=blue):y_8_6:=spacecurve(0,a_8+r*cos(

29、t),b_6+r*sin(t), t=0.2*Pi,color=blue):h00:=plot3d(b0+r*cos(t)*sin(u),a0+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h02:=plot3d(b2+r*cos(t)*sin(u),a0+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h04:=plot3d(b4+r*cos(t)*sin(u),a0+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h06:=plot3d(b6+r

30、*cos(t)*sin(u),a0+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h08:=plot3d(b8+r*cos(t)*sin(u),a0+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h0_2:=plot3d(b_2+r*cos(t)*sin(u),a0+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h0_4:=plot3d(b_4+r*cos(t)*sin(u),a0+r*sin(t),(b_4+r*cos(t)*cos(u),t

31、=0.2*Pi,u=0.2*Pi):h0_6:=plot3d(b_6+r*cos(t)*sin(u),a0+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h0_8:=plot3d(b_8+r*cos(t)*sin(u),a0+r*sin(t),(b_8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h20:=plot3d(b0+r*cos(t)*sin(u),a2+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h22:=plot3d(b2+r*cos(t)*sin(u)

32、,a2+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h24:=plot3d(b4+r*cos(t)*sin(u),a2+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h26:=plot3d(b6+r*cos(t)*sin(u),a2+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h28:=plot3d(b8+r*cos(t)*sin(u),a2+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h2

33、_2:=plot3d(b_2+r*cos(t)*sin(u),a2+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h2_4:=plot3d(b_4+r*cos(t)*sin(u),a2+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h2_6:=plot3d(b_6+r*cos(t)*sin(u),a2+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h2_8:=plot3d(b_8+r*cos(t)*sin(u),a2+r*sin(t),(b

34、_8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h40:=plot3d(b0+r*cos(t)*sin(u),a4+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h42:=plot3d(b2+r*cos(t)*sin(u),a4+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h44:=plot3d(b4+r*cos(t)*sin(u),a4+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h46:=plot3d(b6+r

35、*cos(t)*sin(u),a4+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h48:=plot3d(b8+r*cos(t)*sin(u),a4+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h4_2:=plot3d(b_2+r*cos(t)*sin(u),a4+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h4_4:=plot3d(b_4+r*cos(t)*sin(u),a4+r*sin(t),(b_4+r*cos(t)*cos(u),t

36、=0.2*Pi,u=0.2*Pi):h4_6:=plot3d(b_6+r*cos(t)*sin(u),a4+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h4_8:=plot3d(b_8+r*cos(t)*sin(u),a4+r*sin(t),(b_8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h60:=plot3d(b0+r*cos(t)*sin(u),a6+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h62:=plot3d(b2+r*cos(t)*sin(u)

37、,a6+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h64:=plot3d(b4+r*cos(t)*sin(u),a6+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h66:=plot3d(b6+r*cos(t)*sin(u),a6+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h6_2:=plot3d(b_2+r*cos(t)*sin(u),a6+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi)

38、:h6_4:=plot3d(b_4+r*cos(t)*sin(u),a6+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h6_6:=plot3d(b_6+r*cos(t)*sin(u),a6+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h80:=plot3d(b0+r*cos(t)*sin(u),a8+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h82:=plot3d(b2+r*cos(t)*sin(u),a8+r*sin(t),(b2+

39、r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h84:=plot3d(b4+r*cos(t)*sin(u),a8+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h8_2:=plot3d(b_2+r*cos(t)*sin(u),a8+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h8_4:=plot3d(b_4+r*cos(t)*sin(u),a8+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_20:=plot3d(

40、b0+r*cos(t)*sin(u),a_2+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_22:=plot3d(b2+r*cos(t)*sin(u),a_2+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_24:=plot3d(b4+r*cos(t)*sin(u),a_2+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_26:=plot3d(b6+r*cos(t)*sin(u),a_2+r*sin(t),(b6+r*cos(t)*cos

41、(u),t=0.2*Pi,u=0.2*Pi):h_28:=plot3d(b8+r*cos(t)*sin(u),a_2+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_2_2:=plot3d(b_2+r*cos(t)*sin(u),a_2+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_2_4:=plot3d(b_4+r*cos(t)*sin(u),a_2+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_2_6:=plot3d(b_6+r

42、*cos(t)*sin(u),a_2+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_2_8:=plot3d(b_8+r*cos(t)*sin(u),a_2+r*sin(t),(b_8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_40:=plot3d(b0+r*cos(t)*sin(u),a_4+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_42:=plot3d(b2+r*cos(t)*sin(u),a_4+r*sin(t),(b2+r*cos(t)*cos

43、(u),t=0.2*Pi,u=0.2*Pi):h_44:=plot3d(b4+r*cos(t)*sin(u),a_4+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_46:=plot3d(b6+r*cos(t)*sin(u),a_4+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_48:=plot3d(b8+r*cos(t)*sin(u),a_4+r*sin(t),(b8+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_4_2:=plot3d(b_2+r*cos(t

44、)*sin(u),a_4+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_4_4:=plot3d(b_4+r*cos(t)*sin(u),a_4+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_4_6:=plot3d(b_6+r*cos(t)*sin(u),a_4+r*sin(t),(b_6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_4_8:=plot3d(b_8+r*cos(t)*sin(u),a_4+r*sin(t),(b_8+r*cos(t)*cos

45、(u),t=0.2*Pi,u=0.2*Pi):h_60:=plot3d(b0+r*cos(t)*sin(u),a_6+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_62:=plot3d(b2+r*cos(t)*sin(u),a_6+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_64:=plot3d(b4+r*cos(t)*sin(u),a_6+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_66:=plot3d(b6+r*cos(t)*

46、sin(u),a_6+r*sin(t),(b6+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_6_2:=plot3d(b_2+r*cos(t)*sin(u),a_6+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_6_4:=plot3d(b_4+r*cos(t)*sin(u),a_6+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_6_6:=plot3d(b_6+r*cos(t)*sin(u),a_6+r*sin(t),(b_6+r*cos(t)*cos(u)

47、,t=0.2*Pi,u=0.2*Pi):h_80:=plot3d(b0+r*cos(t)*sin(u),a_8+r*sin(t),(b0+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_82:=plot3d(b2+r*cos(t)*sin(u),a_8+r*sin(t),(b2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_84:=plot3d(b4+r*cos(t)*sin(u),a_8+r*sin(t),(b4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_8_2:=plot3d(b_2+r*cos(t)*s

48、in(u),a_8+r*sin(t),(b_2+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):h_8_4:=plot3d(b_4+r*cos(t)*sin(u),a_8+r*sin(t),(b_4+r*cos(t)*cos(u),t=0.2*Pi,u=0.2*Pi):display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60

49、,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h02,h04,h06,h08,h0_2,h0_4,h0_6,h0_8,h20,h22,h24,h26,h28,h2_2,h2_4,h2_6,h2_8,h40,h42

50、,h44,h46,h48,h4_2,h4_4,h4_6,h4_8,h60,h62,h64,h66,h6_2,h6_4,h6_6,h80,h82,h84,h8_2,h8_4,h_20,h_22,h_24,h_26,h_28,h_2_2,h_2_4,h_2_6,h_2_8,h_40,h_42,h_44,h_46,h_48,h_4_2,h_4_4,h_4_6,h_4_8,h_60,h_62,h_64,h_66,h_6_2,h_6_4,h_6_6,h_80,h_82,h_84,h_8_2,h_8_4,scaling=constrained); July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積

51、計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 20 display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,

52、y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h04,h08,h0_4,h0_8,h22,h26,h2_2,h2_6,h44,h48,h4_4,h4_8,h62,h66,h6_2,h6_6,h80,h84,h8_4,h_20,h_24,h_28,h_2_4,h_2_8,h_44,h_48,h_4_4,h_4_8,h_62,h_66,h_6_2,h_6_6,h_80,h_84,h_8_4,scaling=constrained); July 4,

53、2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 21 display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8

54、,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h08,h0_8,h26,h2_6,h44,h4_4,h62,h6_2,h80,h84,h8_4,h_20,h_28,h_2_8,h_44,h_4_4,h_66,h_6_6,h_80,h_84,h_8_4,scaling=constrained); July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 22d

55、isplay(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8

56、,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h_20,h_28,h_2_8,h_44,h_4_4,h_66,h_6_6,h_80,h_84,h_8_4,scaling=constrained,color=green); July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 23display(xzhou,yzhou,zzhou,yuan,y00,y02,y04,y06,y08,y0_2,y0_4,y0_6,y0_8,y20,y22,y24,y26,

57、y28,y2_2,y2_4,y2_6,y2_8,y40,y42,y44,y46,y48,y4_2,y4_4,y4_6,y4_8,y60,y62,y64,y66,y6_2,y6_4,y6_6,y80,y82,y84,y8_2,y8_4,y_20,y_22,y_24,y_26,y_28,y_2_2,y_2_4,y_2_6,y_2_8,y_40,y_42,y_44,y_46,y_48,y_4_2,y_4_4,y_4_6,y_4_8,y_60,y_62,y_64,y_66,y_6_2,y_6_4,y_6_6,y_80,y_82,y_84,y_8_2,y_8_4,h00,h_20,h_28,h_2_8,

58、h_44,h_4_4,h_66,h_6_6,h_80,h_84,h_8_4,scaling=constrained,color=green); July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 24 July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 25 July 4, 2012四川大學(xué)數(shù)學(xué)學(xué)院 徐小湛旋轉(zhuǎn)體體積計(jì)算公式的幾何意義旋轉(zhuǎn)體體積計(jì)算公式的幾何意義 26 with(plots):xzhou:=spacecurve(x,0,0, x=-2.2, thickness=1,col

59、or=black):yzhou:=spacecurve(0,y,0, y=-2.2, thickness=1,color=black):zzhou:=spacecurve(0,0,z, z=-2.4, thickness=1,color=black):a:=0:b:=3:R:=1:r:=0.1:yuan:=spacecurve(0,a+R*cos(t),b+R*sin(t), t=0.2*Pi, thickness=3,color=red):a0:=0:a2:=0.2:a4:=0.4:a6:=0.6:a8:=0.8:a_2:=-0.2:a_4:=-0.4:a_6:=-0.6:a_8:=-0.8

60、:b0:=3:b2:=3.2:b4:=3.4:b6:=3.6:b8:=3.8:b_2:=3-0.2:b_4:=3-0.4:b_6:=3-0.6:b_8:=3-0.8:y00:=spacecurve(0,a0+r*cos(t),b0+r*sin(t), t=0.2*Pi,color=blue):y02:=spacecurve(0,a0+r*cos(t),b2+r*sin(t), t=0.2*Pi,color=blue):y04:=spacecurve(0,a0+r*cos(t),b4+r*sin(t), t=0.2*Pi,color=blue):y06:=spacecurve(0,a0+r*co