GELE Mathematics — Integral CalculusConcept Map

Concept

Fundamental Theorem of Calculus

Sub Concepts

• Antiderivatives and indefinite integrals
• Definite integrals as limit of Riemann sums
• Connection between F(x) and f(x)

Relationship To Central

Establishes the relationship between differentiation and integration as inverse operations

Concept

Basic Integration Techniques

Sub Concepts

• Power rule integration: ∫x^n dx = x^(n+1)/(n+1) + C
• Logarithmic integration: ∫(1/x)dx = ln|x| + C
• Exponential integration: ∫e^x dx = e^x + C
• Trigonometric integration: ∫sin(x)dx, ∫cos(x)dx
• Substitution method (u-substitution)
• Integration by parts: ∫u dv = uv - ∫v du
• Partial fractions decomposition

Relationship To Central

Core methods for solving indefinite and definite integrals

Concept

Definite Integrals and Area

Sub Concepts

• Evaluation theorem: ∫[a,b] f(x)dx = F(b) - F(a)
• Area under a curve
• Area between two curves: A = ∫[a,b] [f(x) - g(x)]dx
• Properties of definite integrals
• Improper integrals

Relationship To Central

Direct application of integration to calculate accumulated quantities and geometric areas

Concept

Volumes of Revolution

Sub Concepts

• Disk method (about x-axis): V = π∫[a,b] [R(x)]² dx
• Washer method (hollow solids): V = π∫[a,b] ([R(x)]² - [r(x)]²) dx
• Shell method (about y-axis): V = 2π∫[a,b] x·f(x) dx
• Selection of appropriate method based on axis
• Setup and integration procedures

Relationship To Central

Uses integration to calculate volumes of 3D solids formed by rotating 2D curves

Concept

Centroids and Moments

Sub Concepts

• Centroid formula: x̄ = ∫x dA / ∫dA, ȳ = ∫y dA / ∫dA
• Vertical strip element: y_element = y/2
• Horizontal strip element: x_element = x/2
• Moment of inertia: I = ∫y² dA
• Applications in structural analysis and engineering design

Relationship To Central

Uses integration to find geometric centers and mass distribution properties

Concept

Arc Length and Surface Area

Sub Concepts

• Arc length formula: L = ∫√(1 + (dy/dx)²) dx
• Surface area of revolution
• Applications in civil engineering (cable lengths, pipe surfaces)

Relationship To Central

Integration of differential elements to find curve lengths and surface areas

Concept

Common Pitfalls and Board-Exam Mistakes

Sub Concepts

• Forgetting constant of integration (+ C) in indefinite integrals
• Incorrect order in area between curves (upper minus lower)
• Confusion between disk, washer, and shell methods
• Using full radius instead of squared radius in disk method
• Centroid calculation error: using y instead of y/2 for vertical strip
• Incorrect limits of integration
• Sign errors in substitution

Relationship To Central

Critical awareness to avoid errors in PRC examinations

To

Basic Integration Techniques

From

Fundamental Theorem of Calculus

Strength

strong

Relationship

Theorem provides theoretical foundation; techniques are methods to compute integrals

To

Definite Integrals and Area

From

Basic Integration Techniques

Strength

strong

Relationship

Techniques used to evaluate definite integrals for area calculations

To

Volumes of Revolution

From

Definite Integrals and Area

Strength

strong

Relationship

Area formula foundation extended to 3D by slicing and rotating

To

Centroids and Moments

From

Volumes of Revolution

Strength

moderate

Relationship

Both use similar integration setups with differential elements

To

Arc Length and Surface Area

From

Centroids and Moments

Strength

moderate

Relationship

All use integration of differential geometric elements

To

Volumes of Revolution

From

Basic Integration Techniques

Strength

strong

Relationship

Integration techniques required to evaluate volume integrals

To

Centroids and Moments

From

Basic Integration Techniques

Strength

moderate

Relationship

Substitution and algebraic manipulation needed for centroid integrals

To

Centroids and Moments

From

Definite Integrals and Area

Strength

strong

Relationship

Area calculation (denominator) used in centroid formulas

To

Basic Integration Techniques

From

Common Pitfalls and Board-Exam Mistakes

Strength

strong

Relationship

Errors in applying rules (forgetting +C, sign errors in substitution)

To

Volumes of Revolution

From

Common Pitfalls and Board-Exam Mistakes

Strength

strong

Relationship

Confusion between methods, squaring radius incorrectly

To

Centroids and Moments

From

Common Pitfalls and Board-Exam Mistakes

Strength

strong

Relationship

Critical error of using y instead of y/2 for vertical strips

To

Basic Integration Techniques

From

Arc Length and Surface Area

Strength

moderate

Relationship

Power rule and algebraic manipulation needed for arc length integrals

To

NSCP 2015 Structural Design

From

Centroids and Moments

Strength

strong

Relationship

Centroid and moment calculations essential for structural member properties

To

Civil Engineering Applications

From

Volumes of Revolution

Strength

moderate

Relationship

Used for calculating pipe volumes, tank capacities, soil displacement

To

Civil Engineering Applications

From

Definite Integrals and Area

Strength

strong

Relationship

Area calculations foundational for stress analysis, bearing capacity