Area under a curve: Riemann sums

Differentiation was about slopes. Integration is about the opposite kind of question: how much accumulates? The starting point is a concrete picture — the area under a curve.

Fill the area with rectangles

You can't read the area under a curve off a formula the way you'd read a rectangle's. So approximate it with shapes you can measure:

Slice the interval [a, b] into thin strips and stand a rectangle on each. Add up their areas — that total is a Riemann sum. It's only an estimate: the flat tops miss the curve. But make the rectangles thinner and more numerous, and the error shrinks.

From sum to integral

Let the width of each rectangle shrink to zero (so their number grows to infinity). The Riemann sum settles on an exact value — the definite integral:

∫ₐᵇ f(x) dx = lim (n→∞) Σ f(xᵢ) Δx

Read the notation as the picture: the elongated S (∫) is a continuous "sum", f(x) is each rectangle's height, and dx is its infinitely thin width. It's the limits-of-sums idea from pre-calc, made exact.

Signed area

The integral counts area above the x-axis as positive and area below as negative. So ∫ over a full sine hump-and-dip is zero: the areas cancel. "Area under the curve" really means net signed area.

Why this matters

Anything that piles up over an interval is an integral: distance from a velocity, total charge from a current, work from a force. The rectangle picture is why an integral can measure a quantity no simple formula gives you — you sum infinitely many infinitesimal pieces.