Fundamental Theorem of Calculus

The central idea

If $f$ is continuous on $[a,b]$ and $F(x)=\int_a^x f(t)dt$, then $F'(x)=f(x)$ for $a<x<b$ (FTC Part I). If $G$ is any antiderivative of $f$, then $\int_a^b f(x)dx=G(b)-G(a)$ (FTC Part II). The dummy variable $t$ prevents confusion between the integration variable and moving endpoint.

Definitions, hypotheses, and notation

Part I requires enough regularity for the accumulation function to inherit the integrand as its derivative at the point. Continuity is the standard introductory hypothesis. Part II requires an antiderivative across the interval and turns a global accumulation into endpoint data. Constants vanish because $(G(b)+C)-(G(a)+C)=G(b)-G(a)$.

With variable endpoints, orientation explains the signs. Increasing an upper limit adds a thin strip, whereas increasing a lower limit removes one. Thus $d/dx\int_{u(x)}^{v(x)}f(t)dt=f(v(x))v'(x)-f(u(x))u'(x)$. This formula combines FTC Part I, the chain rule, and the convention that reversing integration bounds negates the integral.

Conceptual meaning

Accumulating a continuous rate and then measuring how fast the accumulation changes returns the current rate. Conversely, endpoint subtraction converts infinitely many small contributions into an exact total. The theorem makes differentiation and integration inverse processes, with constants canceled at endpoints.

A dependable method and decision rule

1. Decide whether the task differentiates an accumulation function or evaluates an integral.
2. For a moving upper limit $g(x)$, use $d/dx\int_a^{g(x)}f(t)dt=f(g(x))g'(x)$.
3. For two moving limits, subtract the lower-endpoint contribution.
4. For a definite integral, find an antiderivative and compute upper minus lower.
5. Check continuity or the appropriate weaker integrability assumptions before invoking the theorem.

Fully worked example

Graphical or geometric meaning

For $F(x)=\int_a^x f$, moving the right boundary by a tiny amount $dx$ adds a thin strip with approximate area $f(x)dx$. Dividing by $dx$ leaves $f(x)$, which is the geometric heart of Part I.

Common mistakes and why they fail

Verification and reasonableness checks

• Differentiate any antiderivative used in Part II.
• Estimate sign and magnitude from the integrand's graph.
• For an accumulation function, verify that the derivative has the integrand's output units.

Match the theorem to the endpoints

The evaluation theorem changes $\int_a^b f$ into $F(b)-F(a)$ when $F'=f$. The accumulation theorem differentiates an integral with a moving endpoint. If $G(x)=\int_a^{g(x)}f(t)\,dt$, then $G'(x)=f(g(x))g'(x)$ under the usual continuity assumptions. A moving lower endpoint contributes a minus sign, and two moving endpoints contribute two terms. The dummy variable $t$ separates integration from the outside input. Before differentiating, mark which endpoints move. Before evaluating, check that the antiderivative is valid throughout the interval; discontinuities and improper endpoints require separate treatment rather than an automatic theorem invocation.

Practice

1. Differentiate $\int_0^x e^{t^2}dt$.
2. Evaluate $\int_1^3 2x dx$.
3. Differentiate $\int_x^4\sqrt{1+t^2}dt$.

Connections and next steps

• Antiderivatives
• Definite Integral
• Area Under a Curve