Numerical Integration Formulas

Problem 19.13.

The total mass of a variable density rod is given by 

where m = mass, ρ(x) = density, A_c(x) = cross-sectional area, x = distance along the rod, and L = total length of rod. The following data have been measured for a 20-m (L) length rod. Determine the mass in grams to the best possible accuracy. 

Solution:

First, be aware of the units. Convert the unit for distance into cm to ensure the calculation is dimensionally homogenous. Also, compute the product of density and cross-sectional area of the rod. 

To attain best possible accuracy for the integral, we can apply combination of trapezoidal rule (1st segment), Simpson's 1/3 rule (2nd & 3rd segments) and Simpson's 3/8 rule (last three segments).