Punto 7.docx - Segunda parte(punto 5 al 8 Por medio de las...

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Segunda parte (punto 5 al 8) Por medio de las integrales podemos hallar volúmenes de sólidos de revolución, momentos o centros de masa. 7. Una varilla de longitud 60 cm tiene una densidad lineal que varía proporcionalmente al cuadrado de su distancia a uno de los extremos. Si la densidad en el extremo más pesado es de 7200 g/cm, halle su masa total y el centro de masa. Considere la densidad lineal como: ρ ( x ) = Rx 2 I. Se coloca la varilla sobre el eje x con su extremo más liviano en el origen 0. II. La densidad lineal será ρ ( x ) = Rx 2 , de tal modo que cuando x = 60, P(60) = 7200, luego ρ ( x ) = Rx 2 7200 = R∙ ( 60 ) 2 7200 = R∙ 3600 7200 3600 = R R = 2 Por lo tanto obtenemos que ρ ( x ) = 2 x 2 III. Para hallar la masa evaluamos la integral de la densidad de la siguiente forma: m = 0 60 2 x 2 dx = 2 0 60 x 2 dx = 2 ( x 2 + 1 2 + 1 ) + c = 2 ( x 3 3 ) + c
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m = 0 60 2 x 2 dx = 2 ( x 3 3 ) | 0 60 m = 2 { ( x 3 3 ) ( x 3 3 ) } m = 2 { ( 60 3 3 ) ( 0 3 3 ) } m = 2 { ( 216000 3 ) 0 } m = 2 { 72000 } m = 144000 g Por lo tanto, la masa total de la varilla es 144000 gramos
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