平面绕直线旋转(直线不穿过平面)
二维空间中有区域D,直线AB:
V=2π∫∫r(x,y)dσ V=2\pi \int \int r(x,y)dσV=2π∫∫r(x,y)dσ
不能使用低乘高公式,低乘高有重复,只能用于计算外围的面情况(并且绕x轴旋转的情况是与环状微元的计算方法相容的):
Vx=2π∫∫ydxdy=2π∫ab∫0f(x)ydxdy=π∫abf(x)2dx V_x=2\pi \int \int ydxdy= 2\pi \int_{a}^{b} \int_{0}^{f(x)} ydxdy=\pi \int_{a}^{b} f(x)^2dxVx=2π∫∫ydxdy=2π∫ab∫0f(x)ydxdy=π∫abf(x)2dx
而如果绕y轴转动的话:
Vy=2π∫∫xdxdy=2π∫abxdx∫0f(x)dy=π∫abxf(x)dx V_y= 2\pi \int \int xdxdy= 2\pi \int_{a}^{b} xdx\int_{0}^{f(x)} dy=\pi \int_{a}^{b}x f(x)dxVy=2π∫∫xdxdy=2π∫abxdx∫0f(x)dy=π∫abxf(x)dx