We know how fast he was running. How far did he get? Integration

Speed tells you how fast; integration tells you how far. It's the area under the speed graph, and it's how tracking data turns thousands of speed readings into distance covered and high-speed running.

18 September 2026 Beginner Part 4 of Calculus Through Football

Contents

The football question
The concept
Thinner slices
What tracking data does
Why it matters
Limitations
Try it yourself
Further reading

The football question

We know how fast the winger was running. But how far did he actually get?

Say his speed looked like this (made-up example numbers, from rate of change):

Time	Speed
0 seconds	0.5 m/s
1 second	3.5 m/s
2 seconds	6.5 m/s
3 seconds	8.5 m/s

Distance is speed times time, but his speed never stays still long enough to multiply. So we cheat, one second at a time.

The concept

In the first second he goes from 0.5 to 3.5 m/s, so call it 2 m/s on average, which is 2 metres. The next second averages 5 m/s, so 5 metres. The third averages 7.5, so 7.5 metres. Add them up: 14.5 metres.

That's integration. Plot speed against time and the distance travelled is the area under the curve:

Each strip's area is the distance run in that second: 2 m, then 5 m, then 7.5 m. Together, 14.5 m in three seconds. He covers more in the third second than in the first two together.

Chop time into small slices, work out the little bit of distance in each one, add them together. Written down:

$$\text{distance} = \int_0^3 v(t)\,dt$$

In plain football

v(t) is his speed at time t.
dt is a tiny slice of time; v(t) dt is the tiny bit of distance he covers in it.
∫ from 0 to 3 means: add up every one of those tiny bits from kick-off of the run to three seconds.

Nobody in the pub has ever said "nice definite integral, son", but that's what they're describing.

Thinner slices

With a table of four speeds, averaging each second is the best we can do. Real speed changes smoothly, though, and the thinner the slices, the closer the total gets to the true distance.

Take a smooth sprint from a standing start that levels off at 9.5 m/s (made-up example numbers, the same curve as in the derivative). Slice the first three seconds into rectangles, each using the speed at the start of its slice:

Each rectangle uses the speed at the start of its half-second, so it misses the sliver where he sped up. Thinner slices miss less.

Slice width	Slices	Total
1 second	3	13.08 m
0.5 seconds	6	15.70 m
0.1 seconds	30	17.59 m
0.01 seconds	300	17.99 m
0.001 seconds	3,000	18.03 m

As the slices shrink, the total settles on 18.04 m, the exact area. It's the mirror image of the derivative, where shrinking the gap settled on the acceleration at one moment. Here shrinking the slices settles on the total.

Averaging the start and end of each slice, as we did with the table, gets there much faster: with one-second slices it already gives 17.44 m.

What tracking data does

It's also exactly what tracking data does. The system logs a player's position and speed many times a second. Add them all up and you get the numbers that appear on the telly: total distance covered, high-speed running, sprint distance.

The last two are integrals too, just of part of the curve: add up the distance only while speed is above a line. Take a five-second sprint on the same curve, and split it at 5.5 m/s and 7 m/s (about 20 and 25 km/h; each provider sets its own thresholds):

The same area, split by speed. Of the 36.3 m, most is covered above 7 m/s once he's up to speed; only 3.3 m comes in the slower build-up.

Speed band	Distance
Below 5.5 m/s	3.3 m
5.5 to 7 m/s	3.6 m
Above 7 m/s	29.5 m
Total	36.3 m

So one five-second burst adds 36 m to his distance covered, 33 m to his high-speed running and 29.5 m to his sprint distance. Over a match, those totals are the integrals of thousands of readings.

Why it matters

The derivative asks how quickly something is changing. Integration asks what all that change adds up to. They're two sides of the same idea.
Every "total" in football data is an integral. Distance covered, high-speed running, even an xG timeline, where the chances pile up into a running total through the match.
The slices matter. A GPS unit that samples ten times a second gives a better total than one that samples once a second, for the same reason thinner slices do.
Totals aren't outcomes. Any fan who's watched a team run about daft for 90 minutes, like Celtic on a bad night, and achieve nothing will know those are different questions.

Limitations

Distance covered says nothing about why. A lot of running can mean a team chasing the ball, not controlling the game.
Thresholds are arbitrary. A run at 6.9 m/s doesn't count as a sprint and one at 7.0 does; the numbers jump around the line.
Measurement errors add up too. Small errors in each reading accumulate over thousands of readings, which is why providers filter the raw data.

Try it yourself

Find a match report that gives a player's distance covered. Divide it by the minutes he played to get his average speed in metres per minute, then per second. How does it compare with the 8.5 m/s our winger hit? What does that tell you about how much of a match is walking?