Find the average density ρ_(ave) of the rod.?
1) The linear density ρ in a rod 3 m long is 14/( sqrt of x + 1) kg/m, where x is measured in meters from one end of the rod. Find the average density ρ_(ave) of the rod.
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The average density would be the total weight of the rod divided by the length of the rod, where the total weight would be W= the integral from 0 to 3 of p dx, where p = 14/(sqrt(x+1)). For W we have the integral from 0 to 3 of 14/(sqrt(x+1)) dx = 14/sqrt(u) du where u=x+1. This has the solution 14*2sqrt(u) or 28*sqrt(x+1), which when evaluated from 0 to 3 gives 28*(2 - 1) = 28. So finally the average density will be W/3 = 28/3 = 9.33 kg/m.
Average density of rod = 1/3*â«14/â(x + 1) dx from 0 to 3 = 28/3 kg/m