---
title: "A thin non-uniform rod of length \\(L\\) is positioned along the \\(x\\)-axis from \\(x = 0\\) to \\(x = L\\). The linear mass density of the rod varies quadratically according to the function \\(\\lambda(x) = \\lambda_0\\left(1 + \\dfrac{x^2}{L^2}\\right)\\), where \\(\\lambda_0\\) is a positive constant. Which of the following expressions represents the correct integral setup to determine the center of mass coordinate \\(x_{\\text{cm}}\\) of the rod?"
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url: "https://nerd-notes.com/ubq/120562/"
date_modified: "2026-08-23T04:41:38+00:00"
---

# A thin non-uniform rod of length \(L\) is positioned along the \(x\)-axis from \(x = 0\) to \(x = L\). The linear mass density of the rod varies quadratically according to the function \(\lambda(x) = \lambda_0\left(1 + \dfrac{x^2}{L^2}\right)\), where \(\lambda_0\) is a positive constant. Which of the following expressions represents the correct integral setup to determine the center of mass coordinate \(x_{\text{cm}}\) of the rod?

A thin non-uniform rod of length \(L\) is positioned along the \(x\)-axis from \(x = 0\) to \(x = L\). The linear mass density of the rod varies quadratically according to the function \(\lambda(x) = \lambda_0\left(1 + \dfrac{x^2}{L^2}\right)\), where \(\lambda_0\) is a positive constant. Which of the following expressions represents the correct integral setup to determine the center of mass coordinate \(x_{\text{cm}}\) of the rod?

![A horizontal thin rod lies along an x-axis. The left end of the rod is aligned with a vertical tick mark labeled 0 at the origin, and the right end is aligned with a tick mark labeled L. Below the rod, a horizontal double-headed dimension line spans from 0 to L and is labeled L. Shading inside the rod is light gray at the left end and becomes progressively darker toward the right end to indicate increasing density. A horizontal coordinate axis with an arrow pointing right labeled x runs directly beneath the rod. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460098-ZeO5hT.jpg)

- **A.** \(\dfrac{\int_0^L x \left(1 + \dfrac{x^2}{L^2}\right) dx}{\int_0^L \left(1 + \dfrac{x^2}{L^2}\right) dx}\)
- **B.** \(\dfrac{\int_{-L/2}^{L/2} x \left(1 + \dfrac{x^2}{L^2}\right) dx}{\int_{-L/2}^{L/2} \left(1 + \dfrac{x^2}{L^2}\right) dx}\)
- **C.** \(\dfrac{1}{L} \int_0^L x \left(1 + \dfrac{x^2}{L^2}\right) dx\)
- **D.** \(\dfrac{\int_0^L x^2 \left(1 + \dfrac{x^2}{L^2}\right) dx}{\int_0^L x \left(1 + \dfrac{x^2}{L^2}\right) dx}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120562/*