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Active Modules:
Calculus Curriculum

Calculus: Fundamental Theorems & Applications

Executive Abstract & Intuition

Calculus establishes the quantitative link between instantaneous rates of change and net accumulated quantities. This dual perspective is formalized through the Fundamental Theorem of Calculus.

Theorems & Formulations (3):
1

Part 1: The Accumulation Derivative

Differentiating a continuous accumulation integral retrieves the original integrand function directly.

ddx[∫axf(t) dt]=f(x)\frac{d}{dx} \left[ \int_{a}^{x} f(t)\,dt \right] = f(x)
2

Part 2: Evaluation of Definite Integrals

The net area under a continuous curve equals the antiderivative evaluated at the upper bound minus lower bound.

∫abf(x) dx=F(b)−F(a),where F′(x)=f(x)\int_{a}^{b} f(x)\,dx = F(b) - F(a), \quad \text{where } F'(x) = f(x)
3

Integration by Parts Formula

Derived from the product rule of differentiation, allowing product decomposition into manageable integrals.

∫u dv=uv−∫v du\int u \, dv = u v - \int v \, du

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