The linear density of a rod of length 4 m is given by ρ(x) = 5 + 2(x)^(1/2) measured in kilograms per meter, where x is measured in meters from one end of the rod. Find the total mass of the rod.  

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The mass of the rod is given by the linear density defined in kilograms as linear density*length.

The length of the rod is 4 m and the linear density is varying with `rho(x) = 5 + 2(x)^(1/2)`

The mass of the rod is the definite integral `int_(0)^4 rho(x) dx`

`int_(0)^45 + 2(x)^(1/2) dx`

=> `5x + 2*x^(3/2)/(3/2)` between 0 and 4

=> `5(4 - 0) + (4/3)(4^(3/2) - 0^(3/2))`

=> `20 + (4/3)*(8 - 0)`

=> `20 + 32/3`

=> `92/3`

The mass of the rod is `92/3` kg.

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