Combining Integral Properties

Calculus 1 · Axiom Academy

EXAMPLE Evaluating From Given Values Chain four properties — reversal, additivity, constant multiple, and the difference rule — to evaluate an integral you're only handed in pieces. You are told three facts about two continuous functions f and g — and nothing else: Using only the properties of the definite integral, evaluate . (Notice the middle fact runs its limits from 5 to 2 .) Nicely done — you evaluated a definite integral without ever finding an antiderivative, purely from the properties. Reversal first: a fact given with its limits backwards, , becomes by flipping the sign. Linearity: constants pull out and the difference splits — . Result: 2(7) - 3(6) = 14 - 18 = -4 . The properties turn "unknown functions" into ordinary arithmetic — the everyday way integrals get combined once the Fundamental Theorem is in hand.

This is the written version of the interactive lesson above. See the full Calculus 1 course.