The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) is the result that makes calculus a unified discipline. Without it, computing definite integrals requires evaluating limits of Riemann sums — tedious for every new function. With it, computing $\int_a^b f(x)\,dx$ reduces to evaluating any antiderivative $F$ at two points: $F(b) - F(a)$. This one theorem is the bridge between differentiation (Topic 3) and integration (Topics 5–7).

In finance, the FTC is at work whenever you move from a continuous rate to a total: the total log-return $\int_0^T r(t)\,dt$, the total discounted cash flow $\int_0^T C(t)e^{-rt}\,dt$, the accumulated variance $\int_0^T \sigma^2(t)\,dt$ in stochastic volatility models. Every such computation is an application of FTC Part II.

• state FTC Part I: $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$, and apply it to differentiate area functions;
• state FTC Part II: $\int_a^b f(x)\,dx = F(b) - F(a)$, and use it to evaluate definite integrals;
• explain in words why Part I holds (via the increment-over-$h$ argument);
• apply the chain rule when the upper limit is a composite function (e.g. $\frac{d}{dx}\int_a^{x^2} f(t)\,dt$);
• distinguish FTC Part I (a derivative) from FTC Part II (an evaluation).

1. FTC Part I: differentiating an area function
2. FTC Part II: the evaluation theorem
3. Evaluating definite integrals with FTC Part II
4. The symmetry of the FTC
5. Common errors