Question

设平面薄片所占的闭区域由抛物线y=x^2及直线y=x所围成，它在点(x,y)处的密度μ(x,y)=(x^2)y,求质心

Answer 1

质心定义：x`=(∑μi*xi)/(∑μi), y`=(∑μi*yi)/(∑μi)
积分区域为：0≤x≤1，x^2≤y≤x
x`=(∑μi*xi)/(∑μi)=(∫xμdA)/(∫μdA)
=[∫∫x(x^2)ydxdy]/[∫∫(x^2)ydxdy]
=[∫x(x^2)(∫ydy]dx)/[∫(x^2)(∫ydy)dx]
=[∫x(x^2)(y^2/2)dx]/[∫(x^2)(y^2/2)dx]
=[1/2∫x(x^2)(x^2-x^4)dx]/[1/2∫(x^2)(x^2-x^4)dx] x^2≤y≤x
=[∫(x^5-x^7)dx]/[∫(x^4-x^6)dx]
=(x^6/6-x^8/8)/(x^5/5-x^7/7)
=(1/6-1/8)/(1/5-1/7) 0≤x≤1
=35/48
y`=(∑μi*yi)/(∑μi)=(∫yμdA)/(∫μdA)
=[∫∫y(x^2)ydxdy]/[∫∫(x^2)ydxdy]
=[∫(x^2)(∫y^2dy)dx]/[∫(x^2)(∫ydy)dx]
=[∫(x^2)(y^3/3)dx]/[∫(x^2)(y^2/2)dx]
=[1/3∫(x^2)(x^3-x^6)dx]/[1/2∫(x^2)(x^2-x^4)dx] x^2≤y≤x
=2/3[∫(x^5-x^8)dx]/[∫(x^4-x^6)dx]
=2/3(x^6/6-x^9/9)/(x^5/5-x^7/7)
=2/3(1/6-1/9)/(1/5-1/7) 0≤x≤1
=35/54
∴薄片质心坐标为(x`,y`)=(35/48,35/54)