用二重积分计算旋转体体积的几何解释.ppt

旋转体体积计算公式的几何意义 1,用二重积分计算旋转体体积 的几何解释,龙凤古镇,旋转体体积计算公式的几何意义 2,设D是上半平面内的一个有界闭区域。 将D绕x轴旋转一周得一旋转体，求该旋转体的体积V。,我们用元素法来建立旋转体体积的二重积分公式。,D,旋转体体积计算公式的几何意义 3,D,在区域D的(x,y)处取一个面积元素,它到x轴的距离是 y （如图）。,该面积元素绕x轴旋转而成的旋转体的体积约为：,（体积元素）,于是整个区域绕x轴旋转而成的旋转体的体积为：,旋转体体积计算公式的几何意义 4,D,命题1：上半平面内一个有界闭区域D绕x轴旋转而成的旋转体的体积为：,旋转体体积计算公式的几何意义 5,下面来解释以上公式的几何意义,旋转体体积计算公式的几何意义 6,区域D中一面积元素 绕x轴旋转而成的旋转体为一环形体(如图)。,旋转体体积计算公式的几何意义 7,区域D中一面积元素 绕x轴旋转而成的旋转体为一环形体(如图)。,其体积约为：,（体积元素）,旋转体体积计算公式的几何意义 8,将dV在D上二重积分的几何意义是将划分D的n个面积元素分别绕x轴旋转而成的旋转体相加，得到整个D绕x轴旋转的旋转体。,于是整个区域绕x轴旋转而成的旋转体的体积为：,旋转体体积计算公式的几何意义 9,以下图形给出了这种方法的几何解释,旋转体体积计算公式的几何意义 10,旋转体体积计算公式的几何意义 11,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,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);,旋转体体积计算公式的几何意义 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,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);,旋转体体积计算公式的几何意义 13, 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,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);,旋转体体积计算公式的几何意义 14,旋转体体积计算公式的几何意义 15,设想用电缆做成一个圆环体 那么这个圆环体可由电缆中很多圆环形状的光纤组成 因此，我们可以把这种计算旋转体体积的方法形象地称为 光纤法或电缆法,旋转体体积计算公式的几何意义 16,更多的图形,旋转体体积计算公式的几何意义 17,旋转体体积计算公式的几何意义 18,旋转体体积计算公式的几何意义 19,with(plots): xzhou:=spacecurve(x,0,0, x=-22, thickness=1,color=black): yzhou:=spacecurve(0,y,0, y=-22, thickness=1,color=black): zzhou:=spacecurve(0,0,z, z=-24, 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=02*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: 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=02*Pi,color=blue): y02:=spacecurve(0,a0+r*cos(t),b2+r*sin(t), t=02*Pi,color=blue): y04:=spacecurve(0,a0+r*cos(t),b4+r*sin(t), t=02*Pi,color=blue): y06:=spacecurve(0,a0+r*cos(t),b6+r*sin(t), t=02*Pi,color=blue): y08:=spacecurve(0,a0+r*cos(t),b8+r*sin(t), t=02*Pi,color=blue): y0_2:=spacecurve(0,a0+r*cos(t),b_2+r*sin(t), t=02*Pi,color=blue): y0_4:=spacecurve(0,a0+r*cos(t),b_4+r*sin(t), t=02*Pi,color=blue): y0_6:=spacecurve(0,a0+r*cos(t),b_6+r*sin(t), t=02*Pi,color=blue): y0_8:=spacecurve(0,a0+r*cos(t),b_8+r*sin(t), t=02*Pi,color=blue): y20:=spacecurve(0,a2+r*cos(t),b0+r*sin(t), t=02*Pi,color=blue): y22:=spacecurve(0,a2+r*cos(t),b2+r*sin(t), t=02*Pi,color=blue): y24:=spacecurve(0,a2+r*cos(t),b4+r*sin(t), t=02*Pi,color=blue): y26:=spacecurve(0,a2+r*cos(t),b6+r*sin(t), t=02*Pi,color=blue): y28:=spacecurve(0,a2+r*cos(t),b8+r*sin(t), t=02*Pi,color=blue): y2_2:=spacecurve(0,a2+r*cos(t),b_2+r*sin(t), t=02*Pi,color=blue): y2_4:=spacecurve(0,a2+r*cos(t),b_4+r*sin(t), t=02*Pi,color=blue): y2_6:=spacecurve(0,a2+r*cos(t),b_6+r*sin(t), t=02*Pi,color=blue): y2_8:=spacecurve(0,a2+r*cos(t),b_8+r*sin(t), t=02*Pi,color=blue): y40:=spacecurve(0,a4+r*cos(t),b0+r*sin(t), t=02*Pi,color=blue): y42:=spacecurve(0,a4+r*cos(t),b2+r*sin(t), t=02*Pi,color=blue): y44:=spacecurve(0,a4+r*cos(t),b4+r*sin(t), t=02*Pi,color=blue): y46:=spacecurve(0,a4+r*cos(t),b6+r*sin(t), t=02*Pi,color=blue): y48:=spacecurve(0,a4+r*cos(t),b8+r*sin(t), t=02*Pi,color=blue): y4_2:=spacecurve(0,a4+r*cos(t),b_2+r*sin(t), t=02*Pi,color=blue): y4_4:=spacecurve(0,a4+r*cos(t),b_4+r*sin(t), t=02*Pi,color=blue): y4_6:=spacecurve(0,a4+r*cos(t),b_6+r*sin(t), t=02*Pi,color=blue): y4_8:=spacecurve(0,a4+r*cos(t),b_8+r*sin(t), t=02*Pi,color=blue): y60:=spacecurve(0,a6+r*cos(t),b0+r*sin(t), t=02*Pi,color=blue): y62:=spacecurve(0,a6+r*cos(t),b2+r*sin(t), t=02*Pi,color=blue): y64:=spacecurve(0,a6+r*cos(t),b4+r*sin(t), t=02*Pi,color=blue): y66:=spacecurve(0,a6+r*cos(t),b6+r*sin(t), t=02*Pi,color=blue): y6_2:=spacecurve(0,a6+r*cos(t),b_2+r*sin(t), t=02*Pi,color=blue): y6_4:=spacecurve(0,a6+r*cos(t),b_4+r*sin(t), t=02*Pi,color=blue): y6_6:=spacecurve(0,a6+r*cos(t),b_6+r*sin(t), t=02*Pi,color=blue): y80:=spacecurve(0,a8+r*cos(t),b0+r*sin(t), t=02*Pi,color=blue): y82:=spacecurve(0,a8+r*cos(t),b2+r*sin(t), t=02*Pi,color=blue): y84:=spacecurve(0,a8+r*cos(t),b4+r*sin(t), t=02*Pi,color=blue): y8_2:=spacecurve(0,a8+r*cos(t),b_2+r*sin(t), t=02*Pi,color=blue): y8_4:=spacecurve(0,a8+r*cos(t),b_4+r*sin(t), t=02*Pi,color=blue): y8_6:=spacecurve(0,a8+r*cos(t),b_6+r*sin(t), t=02*Pi,color=blue): y_20:=spacecurve(0,a_2+r*cos(t),b0+r*sin(t), t=02*Pi,color=blue): y_22:=spacecurve(0,a_2+r*cos(t),b2+r*sin(t), t=02*Pi,color=blue): y_24:=spacecurve(0,a_2+r*cos(t),b4+r*sin(t), t=02*Pi,color=blue): y_26:=spacecurve(0,a_2+r*cos(t),b6+r*sin(t), t=02*Pi,color=blue): y_28:=spacecurve(0,a_2+r*cos(t),b8+r*sin(t), t=02*Pi,color=blue): y_2_2:=spacecurve(0,a_2+r*cos(t),b_2+r*sin(t), t=02*Pi,color=blue): y_2_4:=spacecurve(0,a_2+r*cos(t),b_4+r*sin(t), t=02*Pi,color=blue): y_2_6:=spacecurve(0,a_2+r*cos(t),b_6+r*sin(t), t=02*Pi,color=blue): y_2_8:=spacecurve(0,a_2+r*cos(t),b_8+r*sin(t), t=02*Pi,color=blue): y_40:=spacecurve(0,a_4+r*cos(t),b0+r*sin(t), t=02*Pi,color=blue): y_42:=spacecurve(0,a_4+r*cos(t),b2+r*sin(t), t=02*Pi,color=blue): y_44:=spacecurve(0,a_4+r*cos(t),b4+r*sin(t), t=02*Pi,color=blue): y_46:=spacecurve(0,a_4+r*cos(t),b6+r*sin(t), t=02*Pi,color=blue): y_48:=spacecurve(0,a_4+r*cos(t),b8+r*sin(t), t=02*Pi,color=blue): y_4_2:=spacecurve(0,a_4+r*cos(t),b_2+r*sin(t), t=02*Pi,color=blue): y_4_4:=spacecurve(0,a_4+r*cos(t),b_4+r*sin(t), t=02*Pi,color=blue): y_4_6:=spacecurve(0,a_4+r*cos(t),b_6+r*sin(t), t=02*Pi,color=blue): y_4_8:=spacecurve(0,a_4+r*cos(t),b_8+r*sin(t), t=02*Pi,color=blue): y_60:=spacecurve(0,a_6+r*cos(t),b0+r*sin(t), t=02*Pi,color=blue): y_62:=spacecurve(0,a_6+r*cos(t),b2+r*sin(t), t=02*Pi,color=blue): y_64:=spacecurve(0,a_6+r*cos(t),b4+r*sin(t), t=02*Pi,color=blue): y_66:=spacecurve(0,a_6+r*cos(t),b6+r*sin(t), t=02*Pi,color=blue): y_6_2:=spacecurve(0,a_6+r*cos(t),b_2+r*sin(t), t=02*Pi,color=blue): y_6_4:=spacecurve(0,a_6+r*cos(t),b_4+r*sin(t), t=02*Pi,color=blue): y_6_6:=spacecurve(0,a_6+r*cos(t),b_6+r*sin(t), t=02*Pi,color=blue): y_80:=spacecurve(0,a_8+r*cos(t),b0+r*sin(t), t=02*Pi,color=blue): y_82:=spacecurve(0,a_8+r*cos(t),b2+r*sin(t), t=02*Pi,color=blue): y_84:=spacecurve(0,a_8+r*cos(t),b4+r*sin(t), t=02*Pi,color=blue): y_8_2:=spacecurve(0,a_8+r*cos(t),b_2+r*sin(t), t=02*Pi,color=blue): y_8_4:=spacecurve(0,a_8+r*cos(t),b_4+r*sin(t), t=02*Pi,color=blue): y_8_6:=spacecurve(0,a_8+r*cos(t),b_6+r*sin(t), t=02*Pi,color=blue): h00:=plot3d(b0+r*cos(t)*sin(u),a0+r*sin(t),(b0+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h02:=plot3d(b2+r*cos(t)*sin(u),a0+r*sin(t),(b2+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h04:=plot3d(b4+r*cos(t)*sin(u),a0+r*sin(t),(b4+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h06:=plot3d(b6+r*cos(t)*sin(u),a0+r*sin(t),(b6+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h08:=plot3d(b8+r*cos(t)*sin(u),a0+r*sin(t),(b8+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h0_2:=plot3d(b_2+r*cos(t)*sin(u),a0+r*sin(t),(b_2+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h0_4:=plot3d(b_4+r*cos(t)*sin(u),a0+r*sin(t),(b_4+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h0_6:=plot3d(b_6+r*cos(t)*sin(u),a0+r*sin(t),(b_6+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h0_8:=plot3d(b_8+r*cos(t)*sin(u),a0+r*sin(t),(b_8+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h20:=plot3d(b0+r*cos(t)*sin(u),a2+r*sin(t),(b0+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h22:=plot3d(b2+r*cos(t)*sin(u),a2+r*sin(t),(b2+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h24:=plot3d(b4+r*cos(t)*sin(u),a2+r*sin(t),(b4+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h26:=plot3d(b6+r*cos(t)*sin(u),a2+r*sin(t),(b6+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h28:=plot3d(b8+r*cos(t)*sin(u),a2+r*sin(t),(b8+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h2_2:=plot3d(b_2+r*cos(t)*sin(u),a2+r*sin(t),(b_2+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h2_4:=plot3d(b_4+r*cos(t)*sin(u),a2+r*sin(t),(b_4+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h2_6:=plot3d(b_6+r*cos(t)*sin(u),a2+r*sin(t),(b_6+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h2_8:=plot3d(b_8+r*cos(t)*sin(u),a2+r*sin(t),(b_8+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h40:=plot3d(b0+r*cos(t)*sin(u),a4+r*sin(t),(b0+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h42:=plot3d(b2+r*cos(t)*sin(u),a4+r*sin(t),(b2+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h44:=plot3d(b4+r*cos(t)*sin(u),a4+r*sin(t),(b4+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h46:=plot3d(b6+r*cos(t)*sin(u),a4+r*sin(t),(b6+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h48:=plot3d(b8+r*cos(t)*sin(u),a4+r*sin(t),(b8+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h4_2:=plot3d(b_2+r*cos(t)*sin(u),a4+r*sin(t),(b_2+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h4_4:=plot3d(b_4+r*cos(t)*sin(u),a4+r*sin(t),(b_4+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h4_6:=plot3d(b_6+r*cos(t)*sin(u),a4+r*sin(t),(b_6+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h4_8:=plot3d(b_8+r*cos(t)*sin(u),a4+r*sin(t),(b_8+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h60:=plot3d(b0+r*cos(t)*sin(u),a6+r*sin(t),(b0+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h62:=plot3d(b2+r*cos(t)*sin(u),a6+r*sin(t),(b2+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h64:=plot3d(b4+r*cos(t)*sin(u),a6+r*sin(t),(b4+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h66:=plot3d(b6+r*cos(t)*sin(u),a6+r*sin(t),(b6+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h6_2:=plot3d(b_2+r*cos(t)*sin(u),a6+r*sin(t),(b_2+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h6_4:=plot3d(b_4+r*cos(t)*sin(u),a6+r*sin(t),(b_4+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h6_6:=plot3d(b_6+r*cos(t)*sin(u),a6+r*sin(t),(b_6+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h80:=plot3d(b0+r*cos(t)*sin(u),a8+r*sin(t),(b0+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h82:=plot3d(b2+r*cos(t)*sin(u),a8+r*sin(t),(b2+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h84:=plot3d(b4+r*cos(t)*sin(u),a8+r*sin(t),(b4+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h8_2:=plot3d(b_2+r*cos(t)*sin(u),a8+r*sin(t),(b_2+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h8_4:=plot3d(b_4+r*cos(t)*sin(u),a8+r*sin(t),(b_4+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h_20:=plot3d(b0+r*cos(t)*sin(u),a_2+r*sin(t),(b0+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h_22:=plot3d(b2+r*cos(t)*sin(u),a_2+r*sin(t),(b2+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h_24:=plot3d(b4+r*cos(t)*sin(u),a_2+r*sin(t),(b4+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h_26:=plot3d(b6+r*cos(t)*sin(u),a_2+r*sin(t),(b6+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h_28:=plot3d(b8+r*cos(t)*sin(u),a_2+r*sin(t),(b8+r*cos(t)*cos(u),t=02*Pi,u=02*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=02*Pi,u=02*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=02*Pi,u=02*Pi): h_2_6:=plot3d(b_6+r*cos(t)*sin(u),a_2+r*sin(t),(b_6+r*cos(t)*cos(u),t=02*Pi,u=02*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=02*Pi,u=02*Pi): h_40:=plot3d(b0+r*cos(t)*sin(u),a_4+r*sin(t),(b0+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h_42:=plot3d(b2+r*cos(t)*sin(u),a_4+r*sin(t),(b2+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h_44:=plot3d(b4+r*cos(t)*sin(u),a_4+r*sin(t),(b4+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h_46:=plot3d(b6+r*cos(t)*sin(u),a_4+r*sin(t),(b6+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h_48:=plot3d(b8+r*cos(t)*sin(u),a_4+r*sin(t),(b8+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h_4_2:=plot3d(b_2+r*cos(t)*sin(u),a_4+r*sin(t),(b_2+r*cos(t)*cos(u),t=02*Pi,u=02*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=02*Pi,u=02*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=02*Pi,u=02*Pi): h_4_8:=plot3d(b_8+r*cos(t)*sin(u),a_4+r*sin(t),(b_8+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h_60:=plot3d(b0+r*cos(t)*sin(u),a_6+r*sin(t),(b0+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h_62:=plot3d(b2+r*cos(t)*sin(u),a_6+r*sin(t),(b2+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h_64:=plot3d(b4+r*cos(t)*sin(u),a_6+r*sin(t),(b4+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h_66:=plot3d(b6+r*cos(t)*sin(u),a_6+r*sin(t),(b6+r*cos(t)*cos(u),t=02*Pi,u=02*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=02*Pi,u=02*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=02*Pi,u=02*Pi): h_6_6:=plot3d(b_6+r*cos(t)*sin(u),a_6+r*sin(t),(b_6+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h_80:=plot3d(b0+r*cos(t)*sin(u),a_8+r*sin(t),(b0+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h_82:=plot3d(b2+r*cos(t)*sin(u),a_8+r*sin(t),(b2+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h_84:=plot3d(b4+r*cos(t)*sin(u),a_8+r*sin(t),(b4+r*cos(t)*cos(u),t=02*Pi,u=02*Pi): h_8_2:=plot3d(b_2+r*cos(t)*sin(u),a_8+r*sin(t),(b_2+r*cos(t)*cos(u),t=02*Pi,u=02*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=02*Pi,u=02*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,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,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);,旋转体体积计算公式的几何意义 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,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);,旋转体体积计算公式的几何意义 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, 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);,旋转体体积计算公式的几何意义 22,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,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);,旋转体体积计算公式的几何意义 23,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,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);,旋转体体积计算公式的几何意义 24,旋转体体积计算公式的几何意义 25,旋转体体积计算公式的几何意义 26,旋转体体积计算公式的几何意义 27, with(plots): xzhou:=spacecurve(x,0,0, x=-22, thickness=1,color=black): yzhou:=spacecurve(0,y,0, y=-22, thickness=1,color=black): zzhou:=spacecurve(0,0,z, z=-24, 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=02*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: 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=02*Pi,color=blue): y02:=spacecurve(0,a0+r*cos(t),b2+r*sin(t), t=02*Pi,color=blue): y04:=spacecurve(0,a0+r*cos(t),b4+r*sin(t), t=02*Pi,color=blue): y06:=spacecurve(0,a0+r*cos(t),b6+r*sin(t), t=02*Pi,color=blue): y08:=spacecurve(0,a0+r*cos(t),b8+r*sin(t), t=02*Pi,color=blue): y0_2:=spacecurve(0,a0+r*cos(t),b_2+r*sin(t), t=02*Pi,color=blue): y0_4:=spacecurve(0,a0+r*cos(t),b_4+r*sin(t), t=02*Pi,color=blue): y0_6:=spacecurve(0,a0+r*cos(t),b_6+r*sin(t), t=02*Pi,color=blue): y0_8:=spacecurve(0,a0+r*cos(t),b_8+r*sin(t), t=02*Pi,color=blue): y20:=spacecurve(0,a2+r*cos(t),b0+r*sin(t), t=02*Pi,color=blue): y22:=spacecurve(0,a2+r*cos(t),b2+r*sin(t), t=02*Pi,color=blue): y24:=spacecurve(0,a2+r*cos(t),b4+r*sin(t), t=02*Pi,color=blue): y26:=spacecurve(0,a2+r*cos(t),b6+r*sin(t), t=02*Pi,color=blue): y28:=spacecurve(0,a2+r*cos(t),b8+r*sin(t), t=02*Pi,color=blue): y2_2:=spacecurve(0,a2+r*cos(t),b_2+r*sin(t), t=02*Pi,color=bl