Work Done by a Variable Force

The figure shows a graph with force $F(x)$ plotted on the vertical axis and displacement $x$ on the horizontal axis. The curve itself rises from the origin, then gradually flattens out — it represents a force that changes with position, not a constant one. Two vertical lines are drawn at $x_i$ and $x_f$, marking the start and end of the motion we care about.

The key visual is this: the area under the curve between $x_i$ and $x_f$ is approximated by a series of tall, thin rectangles. Each rectangle has width $\Delta x$ and height equal to the force at that particular $x$ — so its area is $F(x) \Delta x$. One of these rectangles is shaded, and next to it the figure labels its area as $\Delta A = F(x) \Delta x$.

What the figure is teaching is the fundamental idea of integration before you have the integral sign. When the force is not constant, you cannot just multiply $F$ by total displacement. Instead, you break the total displacement into many tiny steps $\Delta x$. Over each tiny step, the force is approximately constant, so the work done in that step is $\Delta W \approx F(x) \Delta x$ — exactly the area of that thin rectangle. The total work is then the sum of the areas of all these rectangles.

The textbook then takes the limit as $\Delta x \to 0$, turning the sum of rectangles into the exact area under the curve. That limit is the definite integral:

Here, $W$ is the total work done by the variable force as the object moves from $x_i$ to $x_f$. The symbol $\int$ stands for the integral — the continuous version of the sum. $F(x)$ is the force at each position, and $dx$ is the infinitesimal displacement (the limit of $\Delta x$). The integral literally means "add up $F(x) \cdot dx$ for every tiny step from $x_i$ to $x_f$." …

The figure is a simple but powerful visual: a graph with force $F(x)$ on the vertical axis and position $x$ on the horizontal axis. The curve drawn is some arbitrary function $F(x)$ that changes with $x$ — it is not a straight line, because the force is variable. Two vertical lines are marked at $x_i$ (the initial position) and $x_f$ (the final position). The entire region between the curve, the $x$-axis, and these two vertical boundaries is shaded and labelled "Work".

That shaded area is the central idea. For a constant force, work is just $F \times \Delta x$, which is the area of a rectangle. But when $F$ changes with $x$, you cannot use a single rectangle. The figure shows the textbook's reasoning: if you slice the region under the curve into many thin vertical rectangles of width $\Delta x$, each rectangle has height approximately $F(x)$ at that point. The work done over one such slice is roughly $F(x)\,\Delta x$. Summing all these rectangles gives an approximation to the total work. As $\Delta x$ shrinks to zero — the limit shown in the caption — the sum becomes the exact area under the curve.

This is not a new formula; it is the definition of work for a one-dimensional variable force. The textbook writes it as:

Here $W$ is the work done by the force $F(x)$ as the object moves from $x_i$ to $x_f$. The integral symbol $\int$ is just a shorthand for "sum over infinitesimally thin slices". The expression $F(x)\,dx$ represents the work done over an infinitesimal displacement $dx$, and the integral adds up all those infinitesimal contributions.