Suponha que uma lâmina na forma da região delimitada por y = x-1 e y = 0 ao longo de 1 ≤ x ≤ 4 tem densidade de massa ρ (x,y) = y/x. Qual a massa?

Para calcular a massa total da lâmina, precisamos integrar a densidade de massa ρ(x, y) ao longo da região delimitada por y = x-1 e y = 0, no intervalo de x de 1 a 4. A densidade de massa é dada por ρ(x, y) = y/x. Podemos calcular a massa total usando a integral dupla: M = ∬R ρ(x, y) dA Onde R é a região delimitada por y = x-1 e y = 0, e dA é o elemento de área. Integrando em relação a y primeiro, temos: M = ∫[1,4] ∫[0,x-1] (y/x) dy dx Resolvendo a integral interna em relação a y, temos: M = ∫[1,4] [(1/2)(y^2/x)]|[0,x-1] dx M = ∫[1,4] [(1/2)((x-1)^2/x)] dx M = ∫[1,4] [(1/2)((x^2 - 2x + 1)/x)] dx M = ∫[1,4] [(1/2)(x - 2 + 1/x)] dx M = (1/2) ∫[1,4] (x - 2 + 1/x) dx M = (1/2) [(x^2/2 - 2x + ln|x|)]|[1,4] M = (1/2) [(16/2 - 8 + ln|4|) - (1/2 - 2 + ln|1|)] M = (1/2) [(8 - 8 + ln(4)) - (-1/2 - 2 + ln(1))] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2] M = (1/2) [ln(4) + 5/2]