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, which 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 $F' = f$. The power rule works in reverse for integration:

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

The constant of integration: 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.

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

Economic Areas

The definite integral measures accumulated totals, and in business, those areas represent money. Integration turns rates and margins into totals:

Consumer surplus: The area between the demand curve and the market price $p^*$. Everyone pays $p^*$, but early buyers pay less than their willingness to pay:

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

For linear demand, this equals the triangle area: $CS = \tfrac{1}{2}\,q^*\,(p_{max} - p^*)$. Producer surplus mirrors this below the price line and above supply, measuring total trade value.