🚀 Adventure 6 — Starting Fast, Then Coming to a Stop

Variable acceleration turns area into distance, and turns area into velocity — the visual heart of the Fundamental Theorem of Calculus..

🎯 What you will learn

    Distance traveled = area under v(t).

    Change in velocity = area under a(t).

    Smaller time steps (Δt) give better rectangle estimates — your sums converge.

📖 1) Read / Listen: Story

  • 📖 “Galileo and the Secret of Motion”
  • Follow Galileo’s journey as he discovers how objects speed up and slow down — and how to measure it. You’ll see how he connected the dots between distance, velocity, and acceleration, and how his insights laid the groundwork for calculus.

🎥 2. Watch the Video

3Blue1Brown — Integration and the fundamental theorem of calculus | Chapter 8, Essence of calculus

See how motion builds piece by piece—and then gets “undone” by rates of change. You’ll see something surprising: adding up tiny pieces (of area) and measuring change (derivative) are actually opposite operations — one of the deepest ideas in calculus!

✏️ 3) Do the Adventure 6 Activities

  • ✏️ Activity 1
  • Add up the area under a changing velocity graph to uncover total distance — and see how everything fits together.

  • ✏️ Activity 2
  • Connect what you see: distance from velocity, velocity from acceleration—and watch how area and change fit together perfectly.