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


ρ(x) = {x² if 0≤x≤1 {x(2-x) if 1<x≤2

Verified step by step guidance

1. Identify the density function \( \rho(x) \) given piecewise as \( \rho(x) = x^2 \) for \( 0 \leq x \leq 1 \) and \( \rho(x) = x(2 - x) \) for \( 1 < x \leq 2 \).
2. Recall that the mass \( M \) of a thin bar along the x-axis from \( a \) to \( b \) with density function \( \rho(x) \) is given by the integral \( M = \int_a^b \rho(x) \, dx \).
3. Since the density function is piecewise, split the integral into two parts: \( M = \int_0^1 x^2 \, dx + \int_1^2 x(2 - x) \, dx \).
4. Set up each integral separately: first, \( \int_0^1 x^2 \, dx \), and second, \( \int_1^2 (2x - x^2) \, dx \) after expanding the second density function.
5. Evaluate each integral individually and then add the results to find the total mass of the bar.

Key Concepts

Density Function and Mass

The density function ρ(x) describes how mass is distributed along a one-dimensional object. To find the total mass, you integrate the density over the length of the object, summing the infinitesimal masses at each point.

Piecewise Functions

A piecewise function is defined by different expressions over different intervals. When integrating, you must split the integral at the points where the function changes to correctly calculate quantities like mass.

Definite Integration

Definite integration calculates the accumulation of quantities, such as mass, over an interval. For this problem, integrating the density function over the specified intervals yields the total mass of the bar.