Volume 2: Differential & Integral Calculus

Analytical Overview

Calculus is the mathematical study of continuous change, local linear approximation, and infinitesimal accumulation. In IIT-JEE Advanced and Olympiads, calculus forms approximately 35–40% of the entire examination. This volume provides deep theoretical rigor, $\epsilon$-$\delta$ precision, Leibniz rule mechanics, and geometrical problem-solving techniques.

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Master Chapter Directory

#	Master Chapter	High-Yield JEE Advanced & Olympiad Core
01	Functions, Relations & Functional Equations	Domain/Range determination, Invertibility, Periodicity, Even/Odd symmetry, Cauchy/d'Alembert Functional Equations
02	Limits, Continuity & Differentiability	Indeterminate forms ($0/0,1^{\mathrm{\infty }},\mathrm{\infty }-\mathrm{\infty }$), Squeeze Theorem, Taylor Expansions, Left/Right Differentiability, Differentiability of piecewise & integral functions
03	Application of Derivatives & Monotonicity	Tangents & Normals, Angle between curves, Rolle's & Lagrange Mean Value Theorem, Cauchy MVT, Global Extrema, Concavity & Inflection
04	Indefinite Integration & Standard Forms	Integration by substitution, By parts (LIATE), Partial fractions, Euler substitutions, Reduction formulae
05	Definite Integrals & Leibniz Rule	Fundamental Theorem of Calculus, King's Property, Periodic integral reduction, Leibniz differentiation under integral sign, Walli's formula, Limit of Riemann sum
06	Differential Equations & Area Under Curves	First-order linear differential equations, Bernoulli DE, Exact differentials, Orthogonal trajectories, Area bounded by intersecting curves

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Pedagogical Progression

graph TD
    A["Real Functions & Functional Relations"] --> B["Limits & Continuous Mappings"]
    B --> C["Differential Calculus & Derivatives"]
    C --> D["Extrema, Monotonicity & MVT"]
    C --> E["Riemann Integral & Definite Integration"]
    E --> F["Leibniz Differentiation Under Integral Sign"]
    E --> G["Differential Equations & Analytical Areas"]

Fundamental Theorem of Calculus (FTC)

Let $f: [a, b] \to \mathbb{R}$ be continuous, and define $F(x) = \int_a^x f(t) dt$. Then $F$ is uniformly differentiable on $(a, b)$ with: $$F'(x) = \frac{d}{dx} \left[ \int_a^x f(t) dt \right] = f(x)$$ Furthermore, if $G$ is any antiderivative of $f$ ($G' = f$), then $\int_a^b f(t) dt = G(b) - G(a)$.

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Interactive Calculus Visualizer

Interactive Simulation

Riemann Integral Partition Visualizer

Adjust the number of partitions $N$ and observe how the upper and lower Darboux sums converge to the exact definite integral $\int_a^b f(x)dx$.

Partitions ($N = 12$):Function:f(x) = x² + 1f(x) = sin(x) + 1.2f(x) = 2 · e^{-x²}

Riemann Left Sum:0.0000

Riemann Right Sum:0.0000

Midpoint Approximation:0.0000