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Area Under a Curve in Practice
Definite integrals turn a velocity graph into a distance, and give the area between a curve and the x-axis.
Distance From Velocity
Earlier, differentiating position gave velocity: v() = s'(). Integration reverses this — integrating velocity over a time interval gives the distance travelled in that interval.
Distance travelled from = a to = b = ∫ from a to b of v() dt
Example 1 — distance from a changing velocity
A car's velocity is v() = (metres per second). Find the distance it travels from to .
- Find an antiderivative: F() =
- Evaluate: F() = , F() =
- Subtract:
Answer: metres
Example 2 — a sanity check with constant velocity
A car travels at a constant /s. Find the distance travelled from to using integration, and check it against simple reasoning.
- Integrate: ∫ from to of dt. An antiderivative of the constant is
- Evaluate: F() − F() =
Answer: metres — matching simple reasoning: /s × s = . Integration agrees with ordinary speed × time when speed is constant.
One thing to watch for: this area formula assumes the curve stays above the -axis over the interval. A curve that dips below the -axis contributes a *negative* value to the integral there, which needs extra care beyond this course.
Practice
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