Calculus - Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) Part 1 connects differentiation and integration, showing they are inverse operations. If $f$ is continuous on $[a, b]$, then the function $g(x) = \int_{a}^{x} f(t) dt$ is continuous on $[a, b]$ and differentiable on $(a, b)$, and $g'(x) = f(x)$. Visually, this means the rate of change of the 'area under the curve' from a fixed point $a$ to a moving point $x$ is equal to the height of the function at $x$.

The Fundamental Theorem of Calculus Part 2 (Evaluation Theorem) provides a method for computing definite integrals. It states that $\int_{a}^{b} f(x) dx = F(b) - F(a)$, where $F$ is any antiderivative of $f$ (i.e., $F' = f$). Geometrically, this represents the net signed area bounded by the curve $y = f(x)$, the x-axis, and the vertical lines $x = a$ and $x = b$.

Area Interpretation: In the context of FTC, the definite integral calculates the cumulative area. If the graph of $f(x)$ lies above the x-axis, the integral is the area. If the graph lies below the x-axis, the integral value is negative. The visual representation is a shaded region between the curve and the horizontal axis, bounded by the limits of integration.

Continuity Requirement: For the FTC to be valid, the integrand $f(x)$ must be continuous on the entire closed interval $[a, b]$. If the function has a jump, an asymptote, or a hole within the boundaries, the theorem cannot be applied directly; the integral must instead be split at the point of discontinuity into separate intervals.

The Mean Value Theorem for Integrals: This concept states that for a continuous function $f$ on $[a, b]$, there exists a point $c$ in $(a, b)$ such that $f(c) = \frac{1}{b-a} \int_{a}^{b} f(x) dx$. Visually, this means there is a rectangle with base $(b-a)$ and height $f(c)$ that has exactly the same area as the region under the curve.

Linearity and Additivity: Definite integrals follow the rules of linearity, where $\int [f(x) + g(x)] dx$ is the sum of individual integrals. Additionally, the 'interval addition' property allows us to split the area into two parts at any point $c$: $\int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx$. This is visually equivalent to dividing a single shaded region into two adjacent sub-regions.

Leibniz Rule for Differentiation under the Integral Sign: When the limits of an integral are functions of $x$, say $\int_{u(x)}^{v(x)} f(t) dt$, the derivative is found by substituting the limits into the function and multiplying by their derivatives: $f(v(x)) \cdot v'(x) - f(u(x)) \cdot u'(x)$.