A thin rod occupies the interval [0,L] on the x–axis. Its linear mass density is modeled by the continuous function λ(x)=kx2, where k is a positive constant.
(a) Construct a generic Riemann sum that represents the total mass of the rod using an arbitrary tagged partition of [0,L]. Clearly indicate the physical meaning of each factor in your sum.
(b) Explain why allowing the length of the largest subinterval in your partition to approach 0 yields a unique accumulation of mass. In your explanation, connect the limiting process to the physical idea of slicing the rod into thinner and thinner pieces.
(c) Evaluate the resulting definite integral to obtain an explicit formula for the total mass of the rod in terms of k and L. Show all steps leading to your result.