Calculate and estimate area under a graph

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Calculate and estimate area under a graph

On a velocity-time graph, signed area gives displacement. A triangular region from 0 to 3 seconds rising from 0 to 6 m/s has area (1/2) × 3 × 6 = 9 m. The following rectangle lasts two seconds at 6 m/s and adds 12 m.

When velocity remains nonnegative, distance and displacement agree. With negative velocity, an area below the axis contributes negative displacement; total distance instead uses the magnitudes of the separate areas. Label the required quantity before adding.

For a curved boundary, strips or trapezia give an estimate. For a trapezium of width h with parallel heights a and b, the area is h(a + b)/2. Smaller strips can improve an estimate, but a drawing alone does not prove whether it is an overestimate or underestimate without examining the curve.

Worked example: Find displacement in the illustrated five-second model.

Triangle: 1/2 × 3 × 6 = 9 m.

Rectangle: (5 − 3) × 6 = 12 m.

Add the nonnegative areas.

Answer: 21 m.

Check your understanding: What are the area units when velocity is in m/s and time in s?

Answer: Metres.

Why: Multiplying (m/s) by s cancels the time unit.