Putting Humpty Together Again — What Actually Is an Integral?

By CalculaGames • Updated April 2026

The derivative answers the question:

“How does ___ change as I increase ___?”

This could be anything.

• “How does my position change as I increase time?” → velocity
• “How does the money in my bank account change as time passes?” → income (cashflow rate)

For a more detailed look at derivatives, you can read our article “Golf Courses Everywhere — What Actually Is a Derivative?”.

The integral answers a different question:

“How much total ___ has accumulated as I increase ___?”

Instead of looking at an instantaneous rate of change, we’re looking at the total buildup up to a certain point.

At first glance, these don’t seem very related.

But they are.

Speedometer vs. Odometer

Think about the difference between a speedometer and an odometer.

• The speedometer tells you how your distance is changing at a particular instant.
• The odometer adds up all of those moments and tells you how much total distance you’ve traveled.

In calculus terms:

• Speedometer = derivative of distance
• Odometer = integral of speed

So:

• Derivative: “How is this total changing right now?”
• Integral: “How much total has built up over time?”

Building Totals from Tiny Pieces

Imagine you’re taking a trip from grandma’s house.

At every instant, you move a tiny bit farther away.

If you take a tiny change in distance and divide it by a tiny change in time, you get a very accurate measure of your speed at that moment.

If you make those pieces smaller and smaller, your measurement gets more precise. Push this idea all the way, and you arrive at the derivative.

Now flip the idea around.

If you take all those tiny changes in distance and add them back together, you recover your total distance traveled.

Push that idea to the limit—infinitely many infinitely small pieces—and you get the integral.

• Derivative = chopping into infinitely tiny pieces to measure a rate
• Integral = adding up infinitely tiny pieces to recover a total

Back to the Hill: Accumulating Height

Let’s go back to the golf course.

Now we’re looking at the relationship between:

• horizontal position, and
• vertical height

Here:

• The derivative represents the steepness of the hill
  “How does my elevation change as I move forward?”

That’s slope.

Now for the integral.

As you walk across the hill, you can imagine slicing your path into tiny horizontal steps.

Each step produces a tiny change in height (a small rise or drop).

If you add up all of those tiny rises and drops, you get your total elevation at that point.

Important subtlety:

You’re not directly adding slopes—you’re adding the tiny vertical changes that come from those slopes.

And here’s the beautiful part:

The hill already contains this accumulation.

When you add up all the tiny changes you used to describe it, you simply reconstruct the original hill.

The Big Realization: They Undo Each Other

This works no matter what quantities you’re looking at.

A derivative turns a total into a rate:

• Bank balance → income (rate of change)
• Distance → velocity
• Height → slope

An integral turns a rate back into a total:

• Income → bank balance
• Velocity → distance
• Slope → height

If you know how something changes at every instant—even over incredibly tiny intervals—you can add it all up to recover the full total.

The Fundamental Theorem of Calculus

This relationship has a name:

The Fundamental Theorem of Calculus

It says:

Differentiation and integration are inverse operations.

• Differentiation = slicing into rates
• Integration = summing those slices back into totals

Just like:

• Addition undoes subtraction
• Multiplication undoes division

Integration undoes differentiation.

Even if you don’t yet know how to actually carry out these processes, the core idea is simple and powerful:

Totals slice into rates.
Rates sum into totals.

And those two operations perfectly undo each other.

That’s the beating heart of calculus.