Fundamental Theorem of Calculus

FTC Part 1 (Derivative of an Integral)

If $F(x) = \int_a^x f(t)\,dt$, then: $F'(x) = f(x)$

With the chain rule (variable upper limit): $\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x)) \cdot g'(x)$

FTC Part 2 (Evaluation)

If $F$ is an antiderivative of $f$: $\int_a^b f(x)\,dx = F(b) - F(a)$

Accumulation Functions

$F(x) = \int_a^x f(t)\,dt$ is called an accumulation function.

• $F'(x) = f(x)$ — the integrand is the derivative.
• $F$ is increasing when $f > 0$, decreasing when $f < 0$.
• $F$ has a local max where $f$ changes from positive to negative.

Net Change Theorem

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

The integral of a rate of change gives the net change.

Average Value of a Function

$f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx$

Problem: Let $F(x) = \int_0^{x^2} \sin t\,dt$. Find $F'(x)$.

Solution: By FTC Part 1 with chain rule: $F'(x) = \sin(x^2) \cdot 2x = 2x\sin(x^2)$

• FTC Part 1: $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$.
• FTC Part 2: $\int_a^b f(x)\,dx = F(b) - F(a)$.
• Net change: integral of a rate = total change.
• Average value: $\frac{1}{b-a}\int_a^b f(x)\,dx$.