Separate distance from displacement.

Area between curves integrates upper minus lower; first find their intersections and check which is above on each interval. Total distance integrates speed |v(t)|, while displacement integrates signed velocity v(t). Split a motion interval where velocity changes sign.

When only equally spaced measurements are available, the trapezoidal rule adds areas under straight segments. Interior heights count twice. The result is an approximation unless the actual function is linear over each segment.

INTERACTIVE MODEL

Try it. Watch it change.

f(x)

x^{2}

FunctionUpper limit1Trapezoids8

Integral = 0.3333 · approximation = 0.3359 · total area = 0.3333

Trapezoids approximate the signed integral. Total area adds the magnitudes of regions above and below the axis.

Explore: Select the signed linear function. Compare signed area with total area as the upper limit changes.

Total distance

\int_{t_1}^{t_2}|v(t)|\,\mathrm dt

Area between curves

\int_a^b\bigl(f_{\mathrm{upper}}(x)-f_{\mathrm{lower}}(x)\bigr)\,\mathrm dx

WORKED EXAMPLE

Velocity is v(t) = 2t − 2 m/s for 0 ≤ t ≤ 3. Find displacement and distance.

Displacement = [ $t^{2}$ − 2t]₀³ = 3 m.
The velocity changes sign at t = 1.
Distance = −∫₀¹v dt + ∫₁³v dt = 1 + 4 = 5 m.

Assumed knowledge

Definite integrals and roots of functions.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Find area between curves and integrate rates.
Use a trapezoidal approximation from data.

QCAA Mathematical Methods 2025 v1.3 · p. 28 ↗

READY TO TRY IT?

Separate distance from displacement.

Velocity is v(t) = 2t − 2 m/s for 0 ≤ t ≤ 3. Find displacement and distance.

Put the idea to work.