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 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 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 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 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 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 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 blue 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 sin 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 spacecurve 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 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 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 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 sin 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 sin 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 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 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 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 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 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 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 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 blue 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 sin 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 spacecurve 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 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 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 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 sin t t 0 2 Pi color blue y 28 spacecurve 0 a 2