Work done by variable force — integration: Foundations

Before you can read the parent note Work done by a variable force — integration, you need to own every symbol it throws at you. Below, each one is built from nothing: plain words → a picture → why the topic can't live without it. They are ordered so each brick sits on the one before.

1. Position $x$ — "where am I on the path?"

Picture it. Draw a horizontal line, put a tick at the left marked $0$ . Every point to the right gets a bigger number: $1$ m, $2$ m, $3$ m. That number is $x$ .

Why the topic needs it. A variable force is one whose strength depends on where you are. To even say "the force depends on position", we first need a name for position — that name is $x$ .

Figure — Work done by variable force — integration

2. Displacement $d$ and the tiny step $d x$

Picture it. On the number line, an arrow from $x = 1$ to $x = 3$ has length $2$ : that's a displacement of $2$ m. Now zoom in until the arrow is a hair's width — that microscopic arrow is $d x$ .

Why the topic needs it. The parent note "chops the path into tiny pieces $d x$ ". $d x$ is the width of one sliver.

3. Force $F$ and force-as-a-function $F (x)$

Picture it. Imagine a shopping cart on a floor that gets stickier as you go. At $x = 0$ the needle of a force-meter reads small; at $x = 3$ it reads large. Plotting those readings gives a curve of $F$ against $x$ — the force is different at every spot.

Why the topic needs it. "Variable force" means $F$ is a function of position, $F (x)$ . If $F$ were the same number everywhere it would be constant — see Work done by a constant force.

4. Work $W$ — force spent over distance

Picture it. On the force-vs-position graph, a constant force is a flat horizontal line at height $F$ . Push it over a distance $d$ and the work is the rectangle of height $F$ and width $d$ — area $= F \times d$ .

Why the topic needs it. $W$ is the very quantity we're computing. The parent's mission: get $W$ when the graph's top is no longer flat.

5. Multiplying-and-adding slivers: the sum $\sum$

Picture it. Cut the wiggly area into $N$ thin vertical strips. Strip $i$ has height $F (x_{i})$ (the force there) and width $Δ x_{i}$ (its little width), so its area is $F (x_{i}) Δ x_{i}$ . Adding every strip's area: $W \approx \sum_{i = 1}^{N} F (x_{i}) Δ x_{i}$

Why the topic needs it. This staircase of rectangles is the approximation to the curved area. It's the bridge between "rectangle" and "integral".

6. The limit — "shrink the steps to zero"

Picture it. Few fat strips → the tops of the rectangles poke above/below the curve, area is rough. Many thin strips → the jagged tops smooth into the exact curve. In the limit, the staircase becomes the curve.

Why the topic needs it. The approximation $\approx$ turns into an exact $=$ only in the limit. This is what upgrades a sum into an integral.

7. The integral $\int_{x_{A}}^{x_{B}} F (x) d x$

Picture it. Read the symbol left to right as a sentence: "Add up ( $\int$ ), from start $x_{A}$ to end $x_{B}$ , the quantity height-times-width $F (x) d x$ ." It is the exact area under the curve.

Why the topic needs it. This is the punchline formula of the parent note: $W = \int_{x_{A}}^{x_{B}} F (x) d x$ The maths behind why area = integral is developed in Area under curves and the definite integral.

8. The dot product $F \cdot d r$ and the angle $θ$

Picture it. If you drag a sled by a rope tilted upward, only the forward part of your pull moves the sled; the upward part is wasted (does no work). $cos θ$ measures exactly that forward fraction: $θ = 0^{\circ}$ → $cos θ = 1$ (all of it counts); $θ = 9 0^{\circ}$ → $cos θ = 0$ (none counts, e.g. a normal force); $θ = 18 0^{\circ}$ → $cos θ = - 1$ (force fights the motion, work is negative).

Why the topic needs it. The most general work formula is $W = \int F \cdot d r$ . In 1-D, force and motion align ( $θ = 0$ , $cos θ = 1$ ) so it simplifies to $\int F d x$ — but the $cos θ$ is always secretly there. Full story in Dot product and components of vectors.

How these foundations feed the topic

Equipment checklist

Test yourself — cover the right side and answer before revealing.

What does the symbol

x

stand for?

A number labelling your location along a line, measured from the origin

x = 0

What is the difference between

d

(or

Δ x

) and

d x

d

Δ x

is a finite change in position;

d x

is an infinitely tiny step, small enough that the force barely changes over it.

What does

F (x)

mean in words?

"The force at position

x

" — a rule that gives a force value for each location.

Write the work done by a constant force and say what picture it is.

W = F d

; the rectangle of height

F

and width

d

under the force-vs-position graph.

What does

\sum_{i = 1}^{N} F (x_{i}) Δ x_{i}

represent geometrically?

The total area of

N

thin rectangles approximating the area under the curve.

What happens in the limit

Δ x \to 0

N \to \infty

The staircase of rectangles becomes the exact curved area, and the sum becomes an integral.

Decode each part of

\int_{x_{A}}^{x_{B}} F (x) d x

\int

= sum;

x_{A}, x_{B}

= start and end;

F (x)

= sliver height;

d x

= sliver width.

What does

cos θ

do in

F \cdot d r = F d r cos θ

It keeps only the fraction of the force pointing along the motion;

θ = 9 0^{\circ}

gives zero work,

θ = 18 0^{\circ}

gives negative work.

Why can't

W = F d

handle a variable force?

Because

F

is not one number along the path;

F d

needs a single constant force, so we must integrate instead.

Connections

Work done by a constant force — the flat-line, rectangle case these foundations generalise.
Area under curves and the definite integral — the maths making "integral = area" rigorous.
Dot product and components of vectors — where $cos θ$ and $F \cdot d r$ come from.
Hooke's law and spring potential energy — first place a variable $F (x) = k x$ appears.
Work–Energy Theorem — what all this work actually changes (kinetic energy).
Parent topic — assembles these bricks into $W = \int F d x$ .