7.04c - Areas under graphs

Areas under straight-line segments

At a constant speed of $6\,\mathrm{m/s}$ for $5$ seconds, an object travels $6\times5=30$ metres. On a speed-time graph this is the area of a rectangle of height six and width five. The graph's area is measured in axis-value products, not square centimetres on the paper.

For a speed increasing uniformly from zero, the region is a triangle. Its area, half base times height, is also average speed times duration: $(0+v)/2$ times $t$. A straight segment between two nonzero speeds gives a trapezium, with the two vertical speeds as its parallel sides.

Split a piecewise straight graph at each change of slope. Find every shape's width by subtracting times, then add the areas. Do not multiply the maximum speed by the whole duration unless that speed is maintained throughout.

[DIAGRAM: asset_name: Areas for two motion profiles; asset_slug: polygon-motion-areas; caption: Extending only the final slowing-down stage adds triangular area.; description: Two speed-time graphs have vertices zero zero, four eight, ten eight and respectively fourteen zero or eighteen zero. Each area is split into an initial triangle, middle rectangle and final triangle.]

Change only the final stopping time from fourteen to eighteen seconds. The first two stages stay unchanged.

When all speeds are nonnegative, the area is a distance. The same calculation on a velocity-time graph gives a signed change of position; the distinction matters once the graph goes below the axis.

Use the area method on a fresh profile.

Estimating curved areas

A curved speed graph cannot usually be split into a few exact triangles. Replace short sections by straight chords and calculate the resulting trapezia. The sum estimates the area under the curve.

Use the actual horizontal widths, which need not be equal. For adjacent readings $v_1$ and $v_2$, the strip estimate is $\tfrac12(v_1+v_2)\times\text{width}$. Shared internal readings are used twice, once for each neighbouring strip; they are not extra measurements.

The graph tells you about the direction of error. If every chord lies above a section of curve, those trapezia overestimate its area. If they lie below, they underestimate it. For a curve changing its bend, inspect each strip; a blanket claim about every trapezium estimate is unsafe.

[DIAGRAM: asset_name: Refining a curved-area estimate; asset_slug: curved-area-estimates; caption: Narrower trapezia follow this upward-bending curve more closely.; description: Two panels show the speed curve v equals two plus t squared from zero to four seconds. The first uses two trapezia of width two, and the second four trapezia of width one. Both chord sets lie above the curve between readings.]