The idea in one sentence
An integral adds up a quantity that changes smoothly. Graphically, the integral of a function between two points is the area between its curve and the horizontal axis over that interval. The rest is detail and technique.
Thinking in terms of areas under curves is the friendliest way in. The notation looks formidable, but the core idea is the same as multiplying a rate by a time. The difference is that the rate is allowed to vary.
Start with a rectangle
A vehicle travels at a constant 5 m/s for 4 s. On a graph of speed against time, this is a flat line at height 5 stretching from t = 0 to t = 4.
Distance = speed × time = 5 × 4 = 20 m.
That is also the area of a rectangle 4 wide and 5 tall. This is the key link: on a speed-time graph, the area under the line is the distance travelled. The units agree too, since (m/s) × s gives m.
The same pattern shows up elsewhere. A constant force of 10 N acting over 3 m does 10 × 3 = 30 J of work, which is the area under a flat force-distance graph. A constant flow of 2 litres per minute for 5 minutes delivers 10 litres.
A triangle
Now suppose the speed rises steadily from 0 to 6 m/s over 3 s. The graph is a straight sloping line, and the shape under it is a right triangle with base 3 and height 6.
Area = ½ × base × height = ½ × 3 × 6 = 9.
So the vehicle travels 9 m. A quick check: its average speed is (0 + 6) / 2 = 3 m/s, and 3 m/s for 3 s is 9 m. Both routes agree.
A trapezoid
If the speed rises steadily from 2 to 6 m/s over 4 s, the shape is a trapezoid. Its area is the average of the two parallel sides times the width:
Area = ((2 + 6) / 2) × 4 = 4 × 4 = 16.
The distance is 16 m. Splitting it into a rectangle (2 × 4 = 8) and a triangle (½ × 4 × 4 = 8) gives 8 + 8 = 16, which confirms it.
Curves: slice and add
Real graphs are often curved, so no single standard shape fits. The method is to cut the area into many thin vertical strips. Each strip is almost a rectangle, with area roughly height × width, and adding the strips gives an approximation of the total.
Thinner strips follow the curve more closely. An integral is what you get in the limit as the strips become infinitely thin. That is all the elongated “S” symbol means: a sum of many tiny pieces.
You can do the approximation by hand. For the curve y = x from x = 0 to x = 4, using strips of width 1 and taking the height at the right end of each strip gives 1 + 2 + 3 + 4 = 10. Using the left end gives 0 + 1 + 2 + 3 = 6. The true value lies between them. Geometry says the shape is a triangle with base 4 and height 4, so the exact area is ½ × 4 × 4 = 8, which is indeed between 6 and 10, and the average of the two estimates, 8, happens to match it exactly for a straight line.
The exact answer without strips
For many functions there is a shortcut. If you can find a function whose rate of change is the one you are integrating (an antiderivative), the area between two points is the difference of that function evaluated at them. For y = x, an antiderivative is x²/2.
Between 0 and 4: 4²/2 − 0²/2 = 8 − 0 = 8. This matches the triangle.
For y = 2x between 0 and 3: an antiderivative is x², giving 3² − 0² = 9. The shape is a triangle with base 3 and height 6, and ½ × 3 × 6 = 9. It matches.
The link between rates of change and accumulation is the reason calculus is taught as one subject. Differentiating and integrating are inverse operations.
Signed area
Area below the horizontal axis counts as negative. If a speed graph shows forward motion followed by an equal amount of reverse motion, the total is zero, because the object ends where it started. This is a property of displacement, not distance travelled. If you need the total distance regardless of direction, you add the sizes of the areas.
Where this appears in practice
- Distance from speed, or velocity from acceleration.
- Work from force over distance.
- Charge from current over time. A steady 2 A for 5 s moves 10 C.
- Energy from power over time. A steady 100 W for 60 s delivers 6000 J.
- Volume from a flow rate that changes.
Many instruments and software tools integrate for you, and a numerical method such as summing strips is often what a spreadsheet or an engineering simulation does behind the scenes. Knowing the picture helps you judge whether the result is plausible.
Practical cautions
Check the units on both axes before reading the area. The unit of the area is the product of the two. Make sure the scale on each axis is what you assume, and note that logarithmic axes distort areas. If a graph comes from measurements, remember the numbers carry measurement uncertainty, which carries through to the area.
The examples here use simple illustrative numbers. For anything that affects safety or compliance, follow the methods and standards of your field and have the work reviewed by a qualified person.