Graphical integration: area and slope chains

Slopes give the next derivative, areas give the next integral — read graphs in either direction.

Graphical integration: area and slope chains0 min · Free lecture

In this lesson

  • Slopes give the next derivative, areas give the next integral — read graphs in either direction.

One graph, three stories: slope forward, area backward.

Given any of x(t), v(t), a(t): differentiate (slope) to go down, integrate (area) to go up. Traps hide where the slope is zero (turning points) or the area crosses zero.

Practice the chain: a–t → v–t → x–t, marking where velocity is maximum (a = 0) and where the body turns around (v = 0 with sign change).

Δx=∫v dt,Δv=∫a dt\Delta x = \int v\, dt,\quad \Delta v = \int a\, dt

Graph areas as integrals

Worked example

The slope of a velocity–time graph gives:

  1. Slope = Δv/Δt = acceleration.

Answer: Acceleration

Transcript (0 min)
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Graphical integration: area and slope chains — FemtoLearn.
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Slopes give the next derivative, areas give the next integral — read graphs in either direction.
Printable notes (free)

Practice

Free · 4 questions with full solutions
  1. Q1 · MCQ · difficulty 2/5

    The slope of a velocity–time graph gives:

    • Displacement
    • Acceleration
    • Speed
    • Distance
  2. Q2 · Numerical · difficulty 2/5

    The acceleration of a body starting from rest is a = 2 m/s² for 4 s, then zero for 2 s. Find its velocity at t = 6 s.

  3. Q3 · MCQ · difficulty 2/5

    A particle moves with constant velocity. Which a–t graph describes it?

    • A horizontal line at a = 0
    • A horizontal line at a = 2 m/s²
    • A straight line through the origin
    • A parabola
  4. Q4 · Numerical · difficulty 2/5

    A body moves with v = 10 m/s for 3 s, then v = 5 m/s for 2 s in the same direction. Find the total distance.