# 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.

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...

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.

Approved by eNotes Editorial Team