空间情形下旋转体体积的计算

The Calculation about the Volume of Body of Rotation in Space Situation

运用定积分中的元素法,给出了空间曲线绕空间直线旋转一周所成的旋转曲面与垂直于旋转轴的两个平面所围成的旋转体体积的计算公式:V=π(m2+n2+p2)23∫tt12{[p(y(t)-b)-n(z(t)-c)]2+[m(z(t)-c)-p(x(t)-a)]2+[n(x(t)-a)-m(y(t)-b)]2}m.x′(t)+n.y′(t)+p.z′(t)dt从而将平面图形的旋转体体积推广到了空间情形.

Using of the element method of the definite integral, this paper gives the calculation formula of the volume of the hotly of rotation that encircled hy the surface of revolution which formed by the space curve rotates the space right line a circle and the two planes which perpendicular the rotation axis：V=π/（m^2＋n^2＋p^2）3/2∫^t2 t1{[p（y（t）-b）-n（z（t）-c）]^2＋[m（z（t）-c）-p（x（t）-a）]^2 ＋[n（x（t）-a）-m（y（t）-b）]2}｜m·x′（t）＋n·y＇（t）＋p·z＇（t）｜dt So we extend tile volume of the hody of rotatoin of the plane figure to that of the space situation.