✏️ Adventure 6 — Activity 1 — Area under Velocity Curve
Area under the velocity curve and distance traveled.
Use the velocity graph to estimate total distance by adding rectangle areas,
then compare your estimates with the exact integral value.
Rectangle Table (Δt = 1)
| Interval | Δt | Starting Velocity | vΔt |
|---|---|---|---|
| 0–1 | 1 | ||
| 1–2 | 1 | ||
| 2–3 | 1 | ||
| 3–4 | 1 | ||
| 4–5 | 1 | ||
| 5–6 | 1 | ||
| 6–7 | 1 | ||
| 7–8 | 1 | ||
| Total |
Comparison Table
| Width (Δt) | Method | Sum of Areas |
|---|---|---|
| 1 | Your Estimate | |
| 1/2 | Finer estimate | 85.21 |
| 1/4 | Very fine estimate | 85.31 |
3) Integral Calculation
S = ∫₀⁸ (8t − t²) dt
= [4t² − (1/3)t³]₀⁸
= 256 − 512/3
= 85⅓ ≈ 85.33 m
