Question

设平面薄片所占的闭区域d由抛物线y=x^2及直线y=x所围成,它在点(x,y)处的面密度为ρ(x,y)=xy,求质量

求详细解题步骤

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)