Question

二重积分题目D={(x,y):1≤x≤2,x≤y≤2x}则 ∫∫1/(x+y)^2dxdy=?A.ln2 B.ln2/6

Answer 1

1/(x+y)^2 = (x+y)^(-2) 固定x,对 y 求偏导

∫1/(x+y)^2 dy =- (x+y)^(-1) = -1/(x+y)|_{y=x}^{y=2x} = 1/2x - 1/3x = 1/(6x)

∫1/6x dx = (1/6)ln(x)|_{x=1}^{x=2}=(1/6)ln2 =答案 B.ln2/6

注:_{y=x}^{y=2x} 积分的下限:y=x 上限:y=2x

_{x=1}^{x=2} 积分的下限:x=1 上限:x=2