(1) Masa de piel delgada 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) Coordenadas del centro de masa de la placa delgada:
Abscisa = (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
Coordenada vertical = (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
Corte las coordenadas del centroide (11/10, 2/5)