设L是沿着曲线y=x^3由点(2,8)到(-1,-1),则∫xydx+x^2dy的值

Question

设L是沿着曲线y=x^3由点(2,8)到(-1,-1),则∫xydx+x^2dy的值

Answer 1

这个积分简单,可直接转化为对x的积分,将y=x³代入积分,dy=3x²dx
∫xydx+x²dy
=∫[2-->-1] (xx³+x²*3x²)dx
=∫[2-->-1] 4x⁴dx
=4/5x⁵ [2-->-1]
=-4/5-128/5
=-132/5