繞x軸旋轉(zhuǎn),則旋轉(zhuǎn)半徑為|y|
若焦點在x軸上,設(shè)參數(shù)方程為 x=acost,y=bsint,dx=-asintdt
在橫坐標(biāo)為x處取微元,有dV=πy²dx
∴V=∫(-a,a) πy²dx=∫(-π,0) πb²sin²t*(-asint)dt
=πab²∫(-π,0) (1-cos²t)dcost
=πab²∫(-π,0) dcost-πab²∫(-π,0) cos²tdcost
=πab²cost (-π,0) -1/3*πab²cos³t (-π,0)
=2πab²-2/3*πab²
=4/3*πab²
若焦點在y軸上,設(shè)參數(shù)方程為 x=bcost,y=asint,dx=-bsintdt
在橫坐標(biāo)為x處取微元,有dV=πy²dx
∴V=∫(-b,b) πy²dx=∫(-π,0) πa²sin²t*(-bsint)dt
=πa²b∫(-π,0) (1-cos²t)dcost
=πa²b∫(-π,0) dcost-πa²b∫(-π,0) cos²tdcost
=πa²bcost (-π,0) -1/3*πa²bcos³t (-π,0)
=2πa²b-2/3*πa²b
=4/3*πa²b