The Linear Density Function of the rod

The linear density of a non-homogeneous rod is expressed as ρ(x) = x2 - x + 3 with unit of g/m. If the rod length is 4 m, what is the mass of the rod?

If the rod is divided into many small pieces, the total mass of the rod is the sum of the mass in each piece, which can be expressed as


When the rod is divided into infinite small pieces, the result is the exact mass value.



The Low and Upper Limit of rod

The definite integral definition gives




in which x1 and x2 are the start and the end of the rod. Substituting x1 = 0, x2 = 4, and ρ(x) = x2 - x + 3 into the above formula gives