How do you get distance from a speed graph?

Pick a speed graph, then add rectangles and watch the error fall. Switch between left, right, and midpoint.

Challenge: On the car, sprinter, or bus graph, get within 1% of the exact distance using 8 rectangles or fewer. The box under the picture turns green when you get it.

Stuck? Pick one of the examples from the “Try an example” menu, or press “New example.”

At a steady speed, distance is speed × time. On a speed graph, that's the area of a rectangle: height times width.

When the speed changes, slice the time into $n$ strips of width $\Delta t = T/n$. In each strip, pretend the speed holds steady, and use the height at the left end, the right end, or the middle. Adding the rectangles gives a Riemann sum:

$$\sum_{k=1}^{n} v(t_k)\,\Delta t$$

More, thinner rectangles hug the curve more closely, and the total closes in on the exact area under the curve, the integral:

$$\int_0^{T} v(t)\,dt = \lim_{n \to \infty} \sum_{k=1}^{n} v(t_k)\,\Delta t$$

The small graph on the right shows the error for every number of rectangles from 1 to 50. It drops fast at first, then levels off near zero.

Pick a speed graph, then set the number of rectangles and where each one takes its height.

• The exact distance comes from integrating each formula by hand, so it's exact, not another approximation.
• The error is the rectangle total minus the exact distance, shown as a size.