交换二次积分的积分次序∫0adx∫x√2ax-x2f(x,y)dy=____.
∫0Ady∫a-√a2-y2yf(x,y)dx。解析:因为积分区域为D={(x,y)∣0≤x≤a,x≤y≤√2ax-x2}而曲线y=√2ax-x2为y2=2ax-x2,即(x-a)2+y2=a2是以(a,0)为圆心,a为半径的圆,对于x≤a的半圆的曲线方程为x=a-√a2-y2,所以D={(x,y)∣0≤y≤a,a-√a2-y2≤x≤y}
交换二次积分的积分次序∫0adx∫x√2ax-x2f(x,y)dy=____.
∫0Ady∫a-√a2-y2yf(x,y)dx。解析:因为积分区域为D={(x,y)∣0≤x≤a,x≤y≤√2ax-x2}而曲线y=√2ax-x2为y2=2ax-x2,即(x-a)2+y2=a2是以(a,0)为圆心,a为半径的圆,对于x≤a的半圆的曲线方程为x=a-√a2-y2,所以D={(x,y)∣0≤y≤a,a-√a2-y2≤x≤y}