Mathematics

Problem

A slender metal rod of length 30 centimeters lies along the

x

–axis with its left end at the origin. The rod’s linear mass density, in grams per centimeter, is given by the exponential model

\lambda(x)=5e^{-0.10x}

for

0\le x\le 30

. (a) Set up, but do not evaluate, a definite integral whose value gives the total mass of the rod. (b) Evaluate the integral from part (a) to the nearest tenth of a gram. (c) Verify that the integrand you used in part (a) has units of grams per centimeter and that the result of the integral has units of grams.

Intuitive explanation

The key is recognizing that linear mass density tells you mass per unit length, so the total mass requires integration—you're summing up infinitesimal mass elements $dm = \lambda(x)\,dx$ along the rod. Since density decreases exponentially from left to right, the rod is heaviest near the origin and gets progressively lighter, so most of the mass concentrates at the beginning. Set up the integral $\int_0^{30} \lambda(x)\,dx$ and evaluate it to find the total mass.

Intuitive explanation

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