Skip to content
LearnMathora

Lesson 09 · Core idea two · 9 min

Integrals & area

An integral adds up infinitely many infinitely small pieces. Geometrically: the area under a curve. Practically: the total accumulated by a changing rate.

Build the intuition

Why area, of all things?

Drive at steady speed v for time t and distance is v × t — the area of a rectangle on the speed–time graph. When speed varies, slice time thin: each sliver is nearly-steady, nearly-rectangular. Total distance ≈ total rectangle area. Slice ever thinner and “approximately” sharpens into “exactly”: distance is the area under the speed curve.

The definite integral

The integral of f from a to b is the limit of those rectangle sums as slices shrink — the exact area. The ∫ is a stretched S, for sum; f(x) dx is one infinitely thin slice: height times sliver-width.

∫abf(x) dx\int_a^b f(x)\,dx

Definite vs indefinite

Two cousins share the ∫: the definite integral (with limits a, b) is a number — an area. The indefinite integral (no limits) is a function family — the antiderivative, +C and all. The fundamental theorem, next lesson, is the bridge between them.

Below the axis counts negative

Where the curve dips under the x-axis, accumulated “area” counts negative — driving backward reduces your distance from home. Integrals track net accumulation, signs included.

See it move

InteractiveArea accumulator
4
8
Distance covered from t = 0 to 4: 9.06 — the area under the speed curve. More slices hug the curve more closely.

Approximate with rectangles, then flip to the exact area. The integral is what the rectangles become when the slicing never stops.

A worked example

An integral with no formula at all

  1. Find the area under f(x) = x from 0 to 4.

  2. That region is a triangle: base 4, height 4.

  3. So:

    ∫04x dx=12(4)(4)=8\int_0^4 x\,dx = \tfrac{1}{2}(4)(4) = 8

  4. Integrals are areas first, symbols second — when the region is simple, geometry already knows the answer.

The formula, earned

∫abf(x) dx=net area under f from a to b\int_a^b f(x)\,dx = \text{net area under } f \text{ from } a \text{ to } b

A number: the accumulated total.

∫f(x) dx=F(x)+C\int f(x)\,dx = F(x) + C

A family: every function whose derivative is f.

Out in the world

Your electricity bill

Power consumption rises and falls all day, but you're billed for total energy — the area under your power curve. The utility meter is an integral, spinning faster when the curve runs high.

Common confusion, cleared

“Area under a curve needs a formula for the curve.”

The integral is defined by slicing, not by symbols. Even data with no formula — a sensor log — can be integrated numerically.

“Integrals are always positive (they're areas!).”

Net accumulation can be negative — region below the axis subtracts. Think “running total,” not “paint.”

Check yourself

PracticeQuick check

  1. The area under a speed–time curve gives…

  2. What is ∫₀² 3 dx? (Sketch it.)

Recap

  • Integral = limit of rectangle sums = exact area under the curve.
  • Definite: a number between limits. Indefinite: the antiderivative family.
  • Below the axis counts negative — integrals are net totals.