A thin rod of length meters lies along the -axis from to
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A thin rod of length 2 meters lies along the x-axis from x=0 to x=2. Its linear density (mass per unit length) at a point x meters from the left end is given by λ(x)=kx2, where k is a positive constant measured in kilograms per cubic meter.
Experiments show that the total mass of the rod is 12 kilograms.
(a) Express the mass of the rod as an integral involving λ(x), and evaluate the integral to obtain the mass in terms of k. Do not solve for k in this part.
(b) Use the information that the total mass is 12 kilograms to determine the value of k.
(c) Compute the average linear density of the rod over its entire length. State your answer with appropriate units.
(d) Prove that there exists a point c in the interval (0,2) at which the instantaneous linear density equals the average density found in part (c). Identify that point explicitly.