The change in kinetic energy ($\Delta KE$) of an object is equal to the net work done on it, which can be calculated as the integral of the net force $F(x)$ with respect to $x$:

$\Delta KE = W_{\text{net}} = \int_{x_{\text{initial}}}^{x_{\text{final}}} F(x) \, dx$

If the graph of $F(x)$ vs. $x$ is provided, the change in kinetic energy is represented by the area under the curve of $F(x)$ from $x = 0$ to $x = 3d$.

Expression for $\Delta KE$:

$\Delta KE = \int_{0}^{3d} F(x) \, dx$

This expression is valid regardless of the exact form of $F(x)$. If $F(x)$ is piecewise or follows a specific function, the integral can be computed accordingly.

Would you like assistance interpreting or solving the integral if the function $F(x)$ is specified?

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Questions to Expand Understanding:

1. What is the physical significance of the work-energy theorem in this context?
2. How does the shape of the $F(x)$-vs-$x$ graph influence the kinetic energy change?
3. If $F(x)$ is a constant force, how does the result simplify?
4. How would the calculation change if the motion were not restricted to a horizontal plane?
5. How does this analysis relate to the concept of potential energy?

Tip:

When analyzing force-position graphs, always identify regions with positive and negative force contributions, as they can increase or decrease the total work done.