Differentiation finds "slope (rate of change)," and integration finds "area (accumulated quantity)." They look like completely different questions, but remarkably, the two turn out to be exact opposites of each other. This is the fact behind one of the most important theorems in calculus, the Fundamental Theorem of Calculus (FTC).
To understand it, you need a new concept: the area function. Given a function f(t), A(x) = ∫₀ˣ f(t)dt expresses "the area piled up under f from 0 to x" as a function of x. As x grows, more area piles up, so A(x) grows too. But look at how fast A(x) grows, and something remarkable turns up — A(x)'s instantaneous rate of change (that is, A'(x)) turns out to be exactly the original function f(x)!
Why does this happen? Think about how much the area grows when x increases by just a tiny amount, h, and the clue appears. The newly added sliver of area is close to a very thin rectangle with width h and a height of roughly f(x). So the added area is roughly f(x)×h, and the area's rate of change (change ÷ h) gets close to f(x). Take the limit as h approaches 0, and this approximation becomes an exact equality — proving A'(x)=f(x). This is exactly where differentiation and integration meet.
This fact turns out to be enormously convenient in practice. A function F(x) that becomes f(x) after differentiating is called an antiderivative (or indefinite integral) of f(x), and F(x) turns out to be "siblings," differing only by a constant (C), with the area function A(x) described above. So when computing a definite integral, instead of calculating a limit with the Riemann sum method every single time, you just find an antiderivative F(x) and compute F(b)−F(a). That's the key shortcut that makes computing definite integrals much easier.
One more thing worth remembering: an antiderivative is not just one function, but several. Functions that differentiate to f(x)=2x include x², x²+5, and x²−3. Since differentiating a constant always makes it vanish to 0, adding any constant C still differentiates to the same 2x. That's why an antiderivative is written in the form F(x)+C, representing an entire family of curves shifted up and down from each other.
When studying this with kids, the metaphor "the speed at which area piles up" helps a lot. If you know the rate (inflow) at which water fills a tank, the area under that rate's graph is exactly the amount of water accumulated in the tank. Conversely, if you know how the amount of water changes over time, differentiating that tells you the instantaneous inflow rate. On our activity page, you can move x with a slider to watch area accumulate and confirm that its tangent slope matches the original function, and see how antiderivatives that differ only by a constant C form a family of parallel curves.