---
title: "A particle moves in the xy-plane from the origin (0,0) to the point (x_0, y_0) along the parabolic path y = kx^2, where k, x_0, and y_0 are positive constants such that y_0 = kx_0^2.  During this displacement, the particle is acted upon by a two-dimensional force field given by \\(\\vec{F} = by\\,\\hat{i} + cx\\,\\hat{j}\\), where b and c are non-zero constants with appropriate SI units.  Which of the following expressions correctly represents the integral for the total work done by this force on the particle as it moves along the specified path?"
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url: "https://nerd-notes.com/ubq/120633/"
date_modified: "2026-08-23T04:42:24+00:00"
---

# A particle moves in the xy-plane from the origin (0,0) to the point (x_0, y_0) along the parabolic path y = kx^2, where k, x_0, and y_0 are positive constants such that y_0 = kx_0^2.

During this displacement, the particle is acted upon by a two-dimensional force field given by \(\vec{F} = by\,\hat{i} + cx\,\hat{j}\), where b and c are non-zero constants with appropriate SI units.

Which of the following expressions correctly represents the integral for the total work done by this force on the particle as it moves along the specified path?

A particle moves in the xy-plane from the origin (0,0) to the point (x_0, y_0) along the parabolic path y = kx^2, where k, x_0, and y_0 are positive constants such that y_0 = kx_0^2.

During this displacement, the particle is acted upon by a two-dimensional force field given by \(\vec{F} = by\,\hat{i} + cx\,\hat{j}\), where b and c are non-zero constants with appropriate SI units.

Which of the following expressions correctly represents the integral for the total work done by this force on the particle as it moves along the specified path?

![A Cartesian coordinate system with a horizontal x-axis labeled x and a vertical y-axis labeled y. A solid curved line represents the parabolic path y = k x^2 starting at the origin (0, 0) and curving upward and to the right into the first quadrant to an endpoint labeled (x_0, y_0). A single directional arrowhead along the midpoint of the curve points upward and rightward, indicating motion from the origin toward (x_0, y_0). Dashed projection lines drop from (x_0, y_0) vertically to a tick labeled x_0 on the x-axis and horizontally to a tick labeled y_0 on the y-axis. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460144-pgCiAd.jpg)

- **A.** \(W = \int_0^{x_0} (bkx^2 + cx)\,dx\)
- **B.** \(W = \int_0^{x_0} (b + c)kx^2\,dx\)
- **C.** \(W = \int_0^{x_0} (2b + c)kx^2\,dx\)
- **D.** \(W = \int_0^{x_0} (b + 2c)kx^2\,dx\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120633/*