---
title: "A cart moves along a straight horizontal track, and a sensor records its instantaneous velocity v at equal time intervals of \\(\\Delta t = 2.0\\text{ s}\\), as shown in the table.  | Time \\(t\\text{ (s)}\\) | \\(0.0\\) | \\(2.0\\) | \\(4.0\\) | \\(6.0\\) | |—|—|—|—| | Velocity \\(v(t)\\text{ (m/s)}\\) | \\(3.0\\) | \\(7.0\\) | \\(13.0\\) | \\(21.0\\) |  The velocity of the cart is strictly increasing with non-constant acceleration over the entire time interval from \\(t = 0.0\\text{ s}\\) to \\(t = 6.0\\text{ s}\\). Which of the following correctly identifies the left-endpoint Riemann sum approximation \\(\\Delta x_L\\), the right-endpoint Riemann sum approximation \\(\\Delta x_R\\), and the relationship of the true displacement \\(\\Delta x = \\int_0^6 v(t)\\,dt\\) to these approximations?"
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url: "https://nerd-notes.com/ubq/120492/"
date_modified: "2026-08-23T04:39:09+00:00"
---

# A cart moves along a straight horizontal track, and a sensor records its instantaneous velocity v at equal time intervals of \(\Delta t = 2.0\text{ s}\), as shown in the table.

| Time \(t\text{ (s)}\) | \(0.0\) | \(2.0\) | \(4.0\) | \(6.0\) |
|—|—|—|—|
| Velocity \(v(t)\text{ (m/s)}\) | \(3.0\) | \(7.0\) | \(13.0\) | \(21.0\) |

The velocity of the cart is strictly increasing with non-constant acceleration over the entire time interval from \(t = 0.0\text{ s}\) to \(t = 6.0\text{ s}\). Which of the following correctly identifies the left-endpoint Riemann sum approximation \(\Delta x_L\), the right-endpoint Riemann sum approximation \(\Delta x_R\), and the relationship of the true displacement \(\Delta x = \int_0^6 v(t)\,dt\) to these approximations?

A cart moves along a straight horizontal track, and a sensor records its instantaneous velocity v at equal time intervals of \(\Delta t = 2.0\text{ s}\), as shown in the table.

| Time \(t\text{ (s)}\) | \(0.0\) | \(2.0\) | \(4.0\) | \(6.0\) |
|---|---|---|---|
| Velocity \(v(t)\text{ (m/s)}\) | \(3.0\) | \(7.0\) | \(13.0\) | \(21.0\) |

The velocity of the cart is strictly increasing with non-constant acceleration over the entire time interval from \(t = 0.0\text{ s}\) to \(t = 6.0\text{ s}\). Which of the following correctly identifies the left-endpoint Riemann sum approximation \(\Delta x_L\), the right-endpoint Riemann sum approximation \(\Delta x_R\), and the relationship of the true displacement \(\Delta x = \int_0^6 v(t)\,dt\) to these approximations?

- **A.** | \(\Delta x_L\) | \(\Delta x_R\) | True Displacement \(\Delta x\) | | :---: | :---: | :---: | | \(23\text{ m}\) | \(41\text{ m}\) | \(23\text{ m} < \Delta x < 41\text{ m}\) |
- **B.** | \(\Delta x_L\) | \(\Delta x_R\) | True Displacement \(\Delta x\) | | :---: | :---: | :---: | | \(46\text{ m}\) | \(82\text{ m}\) | \(46\text{ m} < \Delta x < 82\text{ m}\) |
- **C.** | \(\Delta x_L\) | \(\Delta x_R\) | True Displacement \(\Delta x\) | | :---: | :---: | :---: | | \(82\text{ m}\) | \(46\text{ m}\) | \(46\text{ m} < \Delta x < 82\text{ m}\) |
- **D.** | \(\Delta x_L\) | \(\Delta x_R\) | True Displacement \(\Delta x\) | | :---: | :---: | :---: | | \(46\text{ m}\) | \(82\text{ m}\) | \(\Delta x = 64\text{ m}\text{ exactly}\) |

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120492/*