The Accumulation Process

Business Calculus · Axiom Academy

INTRO The Accumulation Process Welcome to Unit 6: Integration for Business What if we know the rate, but want the total? This is the fundamental question that integration answers. So far, we've learned to find rates of change using derivatives: If we know position s(t), we find velocity: v(t) = s'(t) If we know total cost C(x), we find marginal cost: MC = C'(x) What if we know the velocity and want to find position ? What if we know marginal cost and want to find total cost ? This reverse process is called integration , and it's the key to solving accumulation problems in business. You know marginal revenue MR(x). What's the total revenue from selling 100 to 200 units? Marginal cost changes with quantity. What's the total cost of producing the next batch? Profit rate varies over time. What's the total profit over a year? How much total value do consumers gain from buying at the market price? All these problems involve accumulating (adding up) rates to get totals. Integration is the mathematical tool that does this precisely. Here's the key insight: the total accumulation equals the area under the rate curve. If the curve represents a rate (like marginal cost), then: The height at any point is the rate at that quantity The area under the curve is the total accumulation The integral gives us the area = total accumulation Derivatives and integrals are inverse operations - they undo each other! If F'(x) = f(x), then ∫f(x)dx = F(x) + C

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