Adding Motion Up: Feynman on Integration

In Chapter 8-4 of The Feynman Lectures on Physics, Feynman shifts focus from finding velocity given distance (differentiation) to the inverse problem: determining distance from a table of velocities. The key example is a falling ball, but he generalises to a car whose speed varies from moment to moment.

Feynman emphasises that the question is not trivial. A constant speed gives an obvious distance. When speed varies, however, the problem becomes one of reconstructing distance from changing values. This requires a different way of thinking: not averages over long intervals, but the accumulation of small contributions.

He demonstrates that dividing time into intervals Δt, as Δt becomes smaller, the approximation improves. The subtle point is his insistence that this procedure does not immediately give the exact distance. Accuracy depends on how small the intervals are, and the “limit” process is essential to move from approximation to precision.

The integral symbol ∫ is introduced not as an abstraction but as shorthand for this limiting process.

Feynman stresses that the integral is not a new invention but the refinement of summation to infinitesimal scales. The notation is a compressed way of representing this.

Another key point is the inverse relationship: the derivative of an integral is the original function. Differentiation and integration are linked operators, one undoing the other. However, Feynman makes clear that the symmetry is imperfect. Differentiation always yields a definite expression. Integration often does not, at least not in terms of known functions.

Feynman highlights a limitation often overlooked in introductory treatments: while derivatives can always be written algebraically, many integrals cannot. They may resist “closed-form” evaluation and must instead be computed by approximation - the very process of summing over small intervals he began with.

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