The Integral — Accumulating Area
“An integral is 'slice thin and add it all up'.”
The formula
∫ₐᵇ f(x) dx = lim Σ f(xᵢ)·ΔxHow to read it: add up the areas of thin rectangles slicing the interval, then take the limit as their width goes to 0 = the area under the curve
- Δx
- — the width of each thin rectangle
- f(xᵢ)
- — the height of each rectangle
- Σ ... lim
- — sum them all, then take the limit as the width goes to 0
- ∫
- — an elongated S meaning 'sum up'
The hook
The integral sign ∫ looks scary, but it means one thing — 'slice it thin and add it all up'.
In plain words
The area under a curve, filled in with many very narrow rectangles whose areas you add together. Send their width to 0 and you get the exact area.
The intuition
Imagine filling the space under a bumpy roof with thin bricks. Thick bricks give a rough estimate; the thinner you make them, the closer the total hugs the true area. ∫ is that 'infinitely thin sum'.
How it's built
Each slice's area = height f(xᵢ) × width Δx. Add them all up for the Riemann sum Σf(xᵢ)Δx. The limit as the width Δx goes to 0 is the integral.
Example
The area under y=x from x=0 to 2 is a triangle with base 2 and height 2, so ½·2·2 = 2. Accumulating it as an integral gives the same 2.
Common mistake
An integral isn't only 'reverse differentiation'. That's just a handy way to compute it; at heart it's 'the sum of thinly sliced areas'.
Where it's used
Area, volume, distance travelled, accumulated probability, work (energy) — anywhere you need a 'total built up little by little'.
Where it came from
Archimedes was already slicing under curves to find areas (the method of exhaustion). Two thousand years later Newton and Leibniz linked it to differentiation.
Prerequisites
Quick check
What does the integral sign ∫ mean?
- multiply
- slice thin and add up✓
- differentiate
- divide
Practice
What is the area under y=x from x=0 to 4 (a triangle)?
Answer: 8
- A right triangle with base 4 and height 4
- ½·4·4 = 8
Takeaway: Area under a straight line can be double-checked with a geometry formula.
What is the area under y=3 (a constant) from x=0 to 5 (a rectangle)?
Answer: 15
- A rectangle with height 3 and width 5
- 3·5 = 15
Takeaway: Integrating a constant is just a rectangle's area.
Approximate the area under y=x on 0–2 with two width-1 rectangles (right-endpoint heights). Compare with the true area of 2.
Answer: undefined
- Right-endpoint heights: 1 at x=1, 2 at x=2
- Sum of areas = 1·1 + 1·2 = 3
- Bigger than the true 2 — thick slices overestimate
Takeaway: Thick slices give an estimate; thinner slices close in on the truth.