Question

求积分数学题

积分从0到R {根号下[y(2R-y)y]dy}
对不起我打错了.那个Y在根号外面. 是跟号下[y(2R-y)]ydy

Answer 1

[0，R] ∫y√[y(2R-y)]dy=[0，R] {(-1/2)∫√[(2Ry-y²)]d(2Ry-y²)+[0，R]R∫√(2Ry-y²)dy}
=(-1/2)[(2/3)(2Ry-y²)^(3/2)]︱[0，R]+[0，R]R∫√[R²-(R-y)²]dy
=-(1/3)R³+[0，R]R²∫√{1- [1-(y/R)]²}d[1-(y/R)] [将[1-(y/R)]看作下面公式中的u]
=-(1/3)R³+R²{[1-(y/R)]/2}√{1- [1-(y/R)]²}︱[0，R]+(1/2)R²arcsin[1-(y/R)]︱[0，R]
=-(1/3)R³+πR²/4.
其中直接用了公式：∫√(1-u²)du=(u/2)√(1-u²)+(1/2)arcsinu+C

Answer 2

此类积分数学题，关键之处在于要懂得二重积分跟二次积分的转化……一般用图解法，也就是微量元素法，看把哪个量作为积分量最好，也就是看其是x型还是y型！