曲线y=ax^2与y=1-x^2交于点A,直线OA与y=ax^2围成一平面图形,问a为何值时,该图形绕x轴旋转体积最大是

Question

曲线y=ax^2与y=1-x^2交于点A,直线OA与y=ax^2围成一平面图形,问a为何值时,该图形绕x轴旋转体积最大是

Answer 1

容易求得交点为A（1/(a+1)^1/2,a/(a+1))
则V=积分(0,1/(a+1)^1/2) [pi*(R^2-r^2)dx
=pi*积分(0,1/(a+1)^1/2) [a^2*x^2/(a+1)-a^2*x^4]dx
=2*pi*a^2/(15*(a+1)^5/2)
令V'=0
得a=4/5