The Definite Integral

Business Calculus · Axiom Academy

Calculating exact accumulated quantities over an interval 1 What is a Definite Integral? While an indefinite integral gives us a family of functions, a definite integral gives us a specific number - the accumulated total from point a to point b. F(x) is any antiderivative of f(x) Notice there's no "+ C" in a definite integral! The constant cancels out: Answers: "What functions have this derivative?" Answers: "How much accumulates from a to b?" The definite integral _a^b f(x)\,dx represents the signed area between f(x) and the x-axis from x = a to x = b. Area above the x-axis counts as positive. Area below the x-axis counts as negative. The definite integral gives the net area. Example: Calculate _1^4 (2x + 3)\,dx 5 Properties of Definite Integrals Reversing limits changes the sign Integral of a sum is sum of integrals Adjacent intervals can be combined In business, definite integrals answer questions like: _0^ 100 MC(x)\,dx = Cost to produce the first 100 units _0^ 50 MR(x)\,dx = Revenue from selling 50 units _0^ 12 S'(t)\,dt = Total sales over 12 months If f'(x) is the rate of change of f, then _a^b f'(x)\,dx = f(b) - f(a) gives the total change in f from a to b.

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