Question

大学高数题求解

如图

Answer 1

x+y=2 与 y=x 的交点 P(1,1),
(1) 薄皮质量 M = ∫∫u(x,y)dxdy
= ∫dy∫(x+2y)dx
= ∫dy[x^2/2+2yx]
= ∫(2+2y-4y^2)dy = [2y+y^2-4y^3/3] = 5/3.
(2) 薄片质心坐标:
横坐标 = (1/M)∫∫xu(x,y)dxdy
= (3/5)∫dy∫x(x+2y)dx
= (3/5)∫dy[x^3/3+yx^2]
= (3/5)∫[8/3-2y^2-(2/3)y^3]dy
= (3/5)[8y/3-(2/3)y^3-(1/6)y^4]
= (3/5)(11/6)=11/10
纵坐标 = (1/M)∫∫yρ(x,y)dxdy
= (3/5)∫dy∫y(x+2y)dx
= (3/5)∫ydy[x^2/2+2yx]
= (3/5)∫y(2+2y-4y^2)dy
= (3/5)∫y(2+2y-4y^2)dy
= (3/5)[y^2+2y^3/3-y^4] = (3/5)(2/3) = 2/5
薄片质心坐标 (11/10,2/5)