13–20. Mass of one-dimensional objects Find the mass of the following thin bars with the given density function.


ρ(x) = {1 if 0≤x≤2 {2 if 2<x≤3

Verified step by step guidance

1. Identify the density function \( \rho(x) \) given as a piecewise function: \( \rho(x) = 1 \) for \( 0 \leq x \leq 2 \) and \( \rho(x) = 2 \) for \( 2 < x \leq 3 \).
2. Recall that the mass \( m \) of a thin bar along the x-axis with density function \( \rho(x) \) is found by integrating the density over the length of the bar: \( m = \int_a^b \rho(x) \, dx \).
3. Since the density function changes at \( x=2 \), split the integral into two parts: \( m = \int_0^2 1 \, dx + \int_2^3 2 \, dx \).
4. Set up each integral separately: the first integral calculates the mass from 0 to 2 with density 1, and the second integral calculates the mass from 2 to 3 with density 2.
5. Evaluate each integral by integrating the constant densities over their respective intervals, then sum the results to find the total mass of the bar.

Key Concepts

Density Function

The density function ρ(x) describes how mass is distributed along the length of the bar. It can vary with position x, indicating that different segments of the bar may have different mass per unit length. Understanding this function is essential to calculate the total mass accurately.

Definite Integral for Mass

The total mass of a one-dimensional object with variable density is found by integrating the density function over the object's length. This involves computing the definite integral of ρ(x) with respect to x over the given intervals, summing contributions from each segment.

Piecewise Functions

When the density function is defined differently over separate intervals, it is called a piecewise function. To find the total mass, you must integrate each piece over its respective interval and then add the results, ensuring correct handling of the function's discontinuities.