The area of a rectangle or a triangle can be found directly with a formula. But how do you find the area of a shape curved like y=x²? There's no direct formula, so instead we use the idea of approximating it by slicing it into many rectangles. This method is called a Riemann sum — the idea of dividing an interval into pieces and finding the area (a "sum") of each piece.
Say you want to find the area between f(x)=x² and the x-axis, over 0≤x≤2. Slice this interval into n small pieces, and stand up a thin rectangle in each piece with height f(x). Add up the areas of all n rectangles, and you get an approximation of the true area. The problem is that rectangles have sharp corners while a curve is smooth, so they don't fit together perfectly — there's always a slight mismatch.
Here's where the key idea comes in. As you increase the number of rectangles n, each rectangle gets thinner, and the overhanging mismatch with the curve keeps shrinking. Take the limit as n grows without bound, and the mismatch becomes exactly 0, so the sum of the rectangles becomes completely equal to the true area. This limit value is called the definite integral, written ∫₀²x²dx (the ∫ symbol is an old-style S, coming from "Sum"). Notice the concept of a limit playing the key role once again?
The definite integral has one curious property. Integrate a function like f(x)=x, whose graph also dips below the x-axis, and you get a "signed area (net area)" that adds the area above the x-axis and subtracts the area below it. For example, integrating f(x)=x from −1 to 2 gives the triangle area from 0 to 2 (+2) minus the triangle area from −1 to 0 (−0.5), for a result of 1.5. An ordinary shape's area is always positive, but a definite integral's value can come out negative like this, or partially cancel itself out.
A definite integral is extremely useful in real life for calculating an "accumulated amount of change." Integrate a velocity graph, and you get the distance traveled (an interval where the velocity is negative — that is, moving backward — eats into the distance); integrate a stream of income, and you get total income. If you know the rate water flows out of a faucet, you can integrate that rate over a given time span to calculate the total amount of water collected — gathering up an instantaneous rate of change to find the accumulated total amount is exactly what integration does.
When studying this with kids, it's a great idea to have them draw a curve on grid paper themselves and estimate the area by counting small squares. Actually experiencing that the answer gets more accurate the more finely (with smaller grid squares) you count helps them naturally understand "why we take a limit." On our activity page, you can increase the number of rectangles n with a slider and watch directly as the area approaches the exact value, and see with your own eyes how area below the x-axis actually gets subtracted.