Integral Calculus

Adding Up the Margin

If the derivative slices totals into margins, the integral reassembles margins into totals. The two operations undo each other — that is the Fundamental Theorem of Calculus:

$$\int_a^b f(x)\,dx = F(b) - F(a), \qquad F' = f$$

Antiderivatives: $F$ is an antiderivative of $f$ if $F' = f$. Power rule in reverse:

$$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \ne -1)$$

The $+C$ matters in economics: knowing marginal cost pins total cost only up to fixed costs — the constant is the part the margin cannot see.

The definite integral is an area — and business areas are money:

• Marginal cost integrated from 0 to $q$ = total variable cost of producing $q$.
• A sales rate (units/day) integrated over a month = total units sold.
• Marginal profit integrated between two outputs = the profit change between them.

Consumer surplus — the lesson's star application. The demand curve reads as willingness to pay: the first units would have been bought even at high prices. Everyone pays the same market price $p^*$, so buyers who valued the good above $p^*$ pocket the difference. Total windfall:

$$CS = \int_0^{q^*} \big(D(q) - p^*\big)\,dq$$

— the area between the demand curve and the price line. For linear demand it is a triangle: $CS = \tfrac{1}{2}\,q^*\,(p_{max} - p^*)$.

Producer surplus mirrors it below the price line, above supply. Markets create value on both sides of every trade, and the integral is how you measure it in euros rather than adjectives.

Margins reassembled into totals

Common pitfall: Dropping the constant of integration. Marginal cost knows nothing about fixed costs — integrating it recovers total cost only up to $FC$, and forgetting that constant understates every cost estimate by exactly the rent.