MathematicsTHE CONCEPT COLLECTION / 03
Applications of Integrals
Area Under Curves
Choose a strip, identify its two boundaries, and add nonnegative pieces rather than signed cancellation.
THE CONCEPT COLLECTION
Mathematics,
made clearer.
Unpack the concept. Follow the reasoning. Then try it for yourself. A growing reading library for your JEE, NEET, SAT, GRE, GMAT and UPSC common-core preparation.
Independently reviewed topic lessons
489 complete lessons currently available. Each lesson identifies its exact topics within the reviewed analytics or declared teaching framework. Selected past-exam questions and original practice are labeled separately. These notes support learning; they are not complete exam preparation or a claim of current-syllabus coverage.
BROWSE THE INDEX
Mathematics
55 available lessons shown
MathematicsTHE CONCEPT COLLECTION / 03
Sequences and Series
Arithmetic and Geometric Progressions
Recognize a sequence’s structure, then choose pairing, telescoping, a recurrence or a justified limit to sum it.
Topics in this lesson (2)
- Arithmetic and Geometric ProgressionsSequences and Series
- Sequences and SummationsSequences and Series
MathematicsTHE CONCEPT COLLECTION / 03
Binomial Theorem
Binomial Coefficients
Count choices inside an expansion, extract only the needed term, and discard multiples when working modulo an integer.
Topics in this lesson (2)
- Binomial CoefficientsBinomial Theorem
- Remainders and DivisibilityBinomial Theorem
MathematicsTHE CONCEPT COLLECTION / 03
Circles
Circle Geometry and Tangents
Radius, perpendicularity and power: a connected method for circle equations and tangents
MathematicsTHE CONCEPT COLLECTION / 03
Complex Numbers and Quadratic Equations
Complex Number Algebra
Move between algebra, geometry and roots of unity without losing domains or argument branches.
Topics in this lesson (3)
- Complex Number AlgebraComplex Numbers and Quadratic Equations
- Complex Plane GeometryComplex Numbers and Quadratic Equations
- Roots of UnityComplex Numbers and Quadratic Equations
MathematicsTHE CONCEPT COLLECTION / 03
Probability
Conditional Probability and Bayes Theorem
Build the sample space, count with consistent outcomes, and update probabilities when the information changes.
Topics in this lesson (3)
- Conditional Probability and Bayes TheoremProbability
- Probability of EventsProbability
- Counting-Based ProbabilityProbability
MathematicsTHE CONCEPT COLLECTION / 03
Limits Continuity and Differentiability
Continuity and Differentiability
Nearby behavior, matching values and matching slopes: a rigorous route through limits
Topics in this lesson (2)
- Continuity and DifferentiabilityLimits Continuity and Differentiability
- LimitsLimits Continuity and Differentiability
MathematicsTHE CONCEPT COLLECTION / 03
Straight Lines
Coordinate Geometry of Lines and Triangles
Turn geometry into equations: normals, projections, signed area and triangle centers
MathematicsTHE CONCEPT COLLECTION / 03
Integrals
Definite Integrals
Understand the integral as a limit of weighted samples, then use antiderivatives, symmetry and transformations with their hypotheses.
Topics in this lesson (2)
- Definite IntegralsIntegrals
- Integral as a Limit of SumsIntegrals
MathematicsTHE CONCEPT COLLECTION / 03
Statistics
Descriptive Statistics
Summarize the data without hiding its shape: center, spread, transformations and grouped estimates
MathematicsTHE CONCEPT COLLECTION / 03
Matrices and Determinants
Determinants
Use determinant structure to test invertibility, then use the full augmented system to decide consistency.
Topics in this lesson (2)
- DeterminantsMatrices and Determinants
- Linear SystemsMatrices and Determinants
MathematicsTHE CONCEPT COLLECTION / 03
Limits Continuity and Differentiability
Differentiation
Local change with structure: first principles, rules, implicit curves and higher derivatives
MathematicsTHE CONCEPT COLLECTION / 03
Probability
Discrete Random Variables
Turn outcomes into numerical values, then distinguish a fixed count of successes from a waiting time.
Topics in this lesson (2)
- Discrete Random VariablesProbability
- Repeated Independent TrialsProbability
MathematicsTHE CONCEPT COLLECTION / 03
Conic Sections
Ellipse Geometry
A constant distance sum: axes, foci, tangents and affine circle geometry
MathematicsTHE CONCEPT COLLECTION / 03
Sets Relations and Functions
Function Properties and Composition
Domains, reversibility and functional equations: follow the inputs before manipulating the symbols
Topics in this lesson (2)
- Function Properties and CompositionSets Relations and Functions
- Functional EquationsSets Relations and Functions
MathematicsTHE CONCEPT COLLECTION / 03
Trigonometry
Heights and Distances
Build the observation geometry first, then use a complete toolkit of triangle properties
Topics in this lesson (2)
- Heights and DistancesTrigonometry
- Properties of TrianglesTrigonometry
MathematicsTHE CONCEPT COLLECTION / 03
Conic Sections
Hyperbola Geometry
Two branches, a distance difference and asymptotes: keep the signs and exceptional lines visible
MathematicsTHE CONCEPT COLLECTION / 03
Integrals
Indefinite Integrals
Reverse differentiation with a plan: substitution, parts, rational functions and trigonometric structure
MathematicsTHE CONCEPT COLLECTION / 03
Inverse Trigonometric Functions
Inverse Trigonometric Functions
Choose the principal angle first: domains, branches, compositions and inverse-trigonometric equations
MathematicsTHE CONCEPT COLLECTION / 03
Differential Equations
Linear Differential Equations
A unified route from a family of curves to separable, ratio-homogeneous and linear first-order equations.
Topics in this lesson (4)
- Linear Differential EquationsDifferential Equations
- Formation of Differential EquationsDifferential Equations
- Homogeneous Differential EquationsDifferential Equations
- Separable Differential EquationsDifferential Equations
MathematicsTHE CONCEPT COLLECTION / 03
Three Dimensional Geometry
Lines and Planes
Represent directions and normals first, then solve intersections, angles and shortest distances without slope exceptions.
Topics in this lesson (2)
- Lines and PlanesThree Dimensional Geometry
- Distances and Projections in SpaceThree Dimensional Geometry
MathematicsTHE CONCEPT COLLECTION / 03
Logarithms
Logarithms
Exponents read backwards: lawful transformations, changing bases and inequalities with moving bases
MathematicsTHE CONCEPT COLLECTION / 03
Mixed-topic Questions
Equal-weight Cross-chapter Matching
A synthesis workshop: verify each mathematical claim, connect chapters and keep every condition visible
MathematicsTHE CONCEPT COLLECTION / 03
Matrices and Determinants
Matrix Algebra
Track dimensions and multiplication order, then count matrices by their genuinely independent choices.
Topics in this lesson (2)
- Matrix AlgebraMatrices and Determinants
- Counting MatricesMatrices and Determinants
MathematicsTHE CONCEPT COLLECTION / 03
Applications of Derivatives
Maxima and Minima
A derivative suggests where to look. The domain and a comparison decide what is truly largest or smallest.
MathematicsTHE CONCEPT COLLECTION / 03
Conic Sections
Mixed Conic Geometry
Classify before calculating: translations, rotations, degeneracies and intersections across conic families
MathematicsTHE CONCEPT COLLECTION / 03
Applications of Derivatives
Monotonicity and Root Counting
Prove which way a function moves, then count crossings without guessing their locations.
MathematicsTHE CONCEPT COLLECTION / 03
Conic Sections
Parabola Geometry
Equal distances to a focus and a line: parameter points, tangents, normals and focal chords
MathematicsTHE CONCEPT COLLECTION / 03
Complex Numbers and Quadratic Equations
Polynomial Roots
Roots, multiplicities and signs, with a full two-variable quadratic-form extension
Topics in this lesson (2)
- Polynomial RootsComplex Numbers and Quadratic Equations
- Quadratic FormsComplex Numbers and Quadratic Equations
MathematicsTHE CONCEPT COLLECTION / 03
Mathematical Reasoning
Propositional Logic
Truth conditions before intuition: implication, equivalence and valid mathematical arguments
MathematicsTHE CONCEPT COLLECTION / 03
Applications of Derivatives
Related Rates
Differentiate the relationship while the picture is moving; substitute the snapshot only afterward.
MathematicsTHE CONCEPT COLLECTION / 03
Permutations and Combinations
Restricted Counting
Count admissible constructions exactly once, then use prime exponents when factorials become too large to expand.
Topics in this lesson (2)
- Restricted CountingPermutations and Combinations
- Factorials and Prime ExponentsPermutations and Combinations
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Geometry and Trigonometry
Area and volume
Choose the right measure, build a geometric model, and track what changes when a figure is resized.
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Geometry and Trigonometry
Circles
Connect circle geometry with radians, coordinate equations, and the unit circle.
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Advanced Math
Equivalent expressions
Use algebraic structure to reveal the same quantity in a more useful form.
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Algebra
Linear functions
Connect constant rate of change with function notation, tables and contextual models.
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Algebra
Linear inequalities in one or two variables
Describe every allowed value or point, including the boundary and the context.
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Algebra
Linear equations in one variable
Build an equation, preserve equality, and interpret exactly what its solution says.
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Algebra
Systems of two linear equations in two variables
Combine two constraints and understand whether they meet once, never or everywhere.
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Algebra
Linear equations in two variables
Read a line as all ordered pairs satisfying one relationship.
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Geometry and Trigonometry
Lines, angles, and triangles
Turn angle facts, matching vertices, and scale factors into reliable geometric equations.
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Advanced Math
Nonlinear equations in one variable and systems of equations in two variables
Choose a method from the structure, keep domain restrictions, and verify every candidate.
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Advanced Math
Nonlinear functions
Connect shape, formula, table and context for quadratic, exponential and other nonlinear rules.
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Problem-Solving and Data Analysis
One-variable data: Distributions and measures of center and spread
Describe a distribution using its shape, center and spread without losing the raw-data meaning.
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Problem-Solving and Data Analysis
Percentages
Identify the base, use multipliers, and distinguish relative change from percentage points.
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Problem-Solving and Data Analysis
Probability and conditional probability
Count the right outcomes, condition on the right group, and recover missing information.
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Problem-Solving and Data Analysis
Ratios, rates, proportional relationships, and units
Keep quantities proportional and let units guide every conversion.
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Geometry and Trigonometry
Right triangles and trigonometry
Use right-triangle structure, special ratios, and angle-based relationships to find exact or approximate measurements.
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Problem-Solving and Data Analysis
Inference from sample statistics and margin of error
Use a representative sample to estimate a population and express the uncertainty honestly.
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Problem-Solving and Data Analysis
Evaluating statistical claims: Observational studies and experiments
Separate how people were selected from how treatments were assigned, then limit the claim accordingly.
MathematicsTHE CONCEPT COLLECTION / 03
SAT · Problem-Solving and Data Analysis
Two-variable data: Models and scatterplots
Read paired data, select a defensible model, and distinguish predictions from observations.
MathematicsTHE CONCEPT COLLECTION / 03
Sets Relations and Functions
Sets and Relations
From membership to structure: set algebra, ordered pairs and equivalence classes
MathematicsTHE CONCEPT COLLECTION / 03
Applications of Derivatives
Tangents and Normals
Use the local direction of a curve without losing vertical lines, singular points or the point of contact.
MathematicsTHE CONCEPT COLLECTION / 03
Trigonometry
Trigonometric Identities
Transform with purpose, solve with complete periods: identities and equations with domain control
Topics in this lesson (2)
- Trigonometric IdentitiesTrigonometry
- Trigonometric EquationsTrigonometry
MathematicsTHE CONCEPT COLLECTION / 03
Vector Algebra
Vector Products and Geometry
Use displacements, projections and oriented areas to turn spatial geometry into controlled algebra.
Topics in this lesson (3)
- Vector Products and GeometryVector Algebra
- Position Vectors and Section FormulaVector Algebra
- Spatial Areas and VolumesThree Dimensional Geometry
No available notes match. Try another topic or chapter.
LET CURIOSITY LEAD. USE DATA FOR CONTEXT.
From a pattern to a plan.
Explore reviewed historical question frequencies, the SAT blueprint and practice mix, or GRE and GMAT structure and sample evidence, then open chapter notes or jump directly to a topic lesson.