Introduction to Integration

Derivatives cut a function ever finer to inspect the slope at each point; integration runs the other way — it adds countless small pieces back into a whole. How can infinitely many pieces sum to one definite answer? Exactly the "approaching without end" idea from Limits.

Slice the area into strips

Find the area under $y = x$ from 0 to 2. It is a triangle: base 2, height 2, area $\frac{1}{2} \times 2 \times 2 = 2$ .

Even without the triangle formula, strips close in on the answer:

Worked example: rectangles boxing the area

Cover the region with two rectangles of width 1. Taking heights at left endpoints gives $0 \times 1 + 1 \times 1 = 1$ , too small; at right endpoints, $1 \times 1 + 2 \times 1 = 3$ , too big. The true answer 2 sits neatly between. The thinner the strips, the closer both estimates hug 2 — sliced infinitely thin, they "equal" 2.

Slice infinitely fine, then add everything up — that move has an official name: integration.

The ∫ sign and the anti-derivative

The integral sign $\int$ is a stretched S, a reminder that its heart is Summing. Its most common use asks: whose derivative is the function in my hand?

$\int 2x \, dx = x^2 + C$

Since $(x^2)' = 2x$ , the integral of 2x is $x^2$ . Integration is the reverse of differentiation: what the derivative took apart, the integral puts back. The extra $C$ is the constant of integration — $x^2 + 5$ and $x^2 - 3$ both differentiate to 2x, and C bookkeeps the difference when reassembling.

The definite integral: adding bounds

Write the range above and below the integral sign, and the answer turns from a family of functions into a single number:

$\int_0^2 x \, dx = \left[ \frac{x^2}{2} \right]_0^2 = \frac{4}{2} - 0 = 2$

Exactly the triangle area from the first section! With bounds it is a definite integral (a number, usually an area or a total); without bounds an indefinite integral (a family of functions, trailing a C).

What integration is for

Distance from speed: cruising at 30 km/h for 2 hours, the area under the speed-time graph is a rectangle worth $30 \times 2 = 60$ km — area under the graph is distance traveled. A changing speed works the same way, only with curvy-topped pieces;
Totals from flow rates: water flowing at 5 liters per minute delivers 50 liters in 10 minutes; when the flow surges and dips, add up each small stretch's contribution — integration again;
Meter readings, probability under a curve — wherever "something changing must be accumulated", integration is behind it.

Worked example: speed to distance

A car cruises at 30 km/h for 2 hours: distance = area under the speed graph = $30 \times 2 = 60$ km. If it drives 20 km/h for the first hour and 40 km/h for the second, the area becomes two rectangles: $20 \times 1 + 40 \times 1 = 60$ km — slice first, then add.

Guess, then verify

Before integrating, ask whose derivative this is. Guess a candidate, then check by differentiating — this habit of "guess and verify" settles most integrals you meet as a beginner.

Quick quiz

1. Under y = x from 0 to 2 sits a triangle. What is its area?
2. Which function has derivative 2x?
3. Driving steadily at 30 km/h for 2 hours: the area under the speed graph is the distance. How far?