Integral Calculus

• Topic: Area under the graph
• Introduction to integral calculus and its applications
• Definition of the definite integral
• Fundamental Theorem of Calculus
• Notations used in integral calculus

Area under the graph

• Finding the area of a region bounded by a curve and the x-axis
• Example: Find the area under the curve y = f(x) between x = a and x = b
• Techniques to find the area under the graph
  • Using basic geometric shapes (rectangles, triangles)
  • Using algebraic methods (integration)
• Importance of finding the area under the graph

Definite integral

• Definition of a definite integral
• Interpretation of the definite integral as the area under the curve
• Notation: ∫[a, b] f(x) dx
• Properties of definite integrals
  • Linearity property
  • Change of limits property
  • Comparison property

Example: Finding the definite integral

• Given function f(x) = x^2 + 3x - 2
• Find the value of ∫[-1, 2] f(x) dx
• Step-by-step solution using antiderivative and definite integral

Fundamental Theorem of Calculus

• Statement of the Fundamental Theorem of Calculus
• Part 1: Evaluation of definite integrals using antiderivatives
• Part 2: Derivative of an integral function
• Application of the Fundamental Theorem of Calculus in solving problems

Techniques of integration

• Substitution method: u-substitution
  • Example: Evaluate ∫ (2x + 3)^4 dx using u-substitution
• Integration by parts
• Partial fractions
• Trigonometric integrals
• Rationalization techniques

Example: Application of integration techniques

• Given function f(x) = 1/(x^2 - x - 2)
• Find the indefinite integral ∫ f(x) dx
• Step-by-step solution using partial fractions and integration techniques

Integration of trigonometric functions

• Basic trigonometric identities
• Integration of basic trigonometric functions
  • Example: ∫ sin^3(x) cos^2(x) dx
• Integration of inverse trigonometric functions
  • Example: ∫ sec^2(x) dx

Applications of integral calculus

• Calculation of areas and volumes
• Finding the center of mass
• Calculating work and fluid pressure
• Evaluating definite integrals representing probability distributions
• Application in physics, engineering, economics, and other fields

Summary

• Recap of key concepts covered
• Importance of integral calculus in various fields
• Practice problems to reinforce understanding
• Encouragement to explore more applications and examples
• Next topics in the syllabus

Integral Calculus - Example on area under graph

• Given function: f(x) = 2x + 3
• Find the area under the graph of f(x) between x = 1 and x = 5
• Step 1: Plot the graph of f(x) = 2x + 3
• Step 2: Identify the region bounded by the graph and the x-axis
• Step 3: Break the region into smaller sections
• Step 4: Calculate the area of each section
• Step 5: Add the areas to find the total area under the graph

Area under graph - Step 1

• Plotting the graph of f(x) = 2x + 3
• x-axis: -2 to 6, y-axis: -1 to 15
• Marking the points (1, 5) and (5, 13) on the graph

Area under graph - Step 2

• Identifying the region bounded by the graph and the x-axis
• The region is a trapezoid with the x-axis as the base
• The higher side of the trapezoid is the graph of f(x) = 2x + 3

Area under graph - Step 3

• Breaking the region into smaller sections
• Section 1: Rectangle with base (1, 3) to (2, 7)
• Section 2: Trapezoid with vertices (2, 7), (3, 9), (4, 11), (5, 13)

Area under graph - Step 4

• Calculating the area of each section
• Area of section 1: (2-1) * (7-3) = 4 square units
• Area of section 2: (5-4) * [(9+11)/2] = 10 square units

Area under graph - Step 5

• Adding the areas to find the total area
• Total area = Area of section 1 + Area of section 2
• Total area = 4 + 10 = 14 square units

Integration

• Introduction to the concept of integration
• Definition of integration as the reverse process of differentiation
• Notation: ∫ f(x) dx
• Evaluating integrals using antiderivatives
• Examples of evaluating integrals

Integrating basic functions

• Integration of constant functions
  • Example: ∫ 3 dx
• Integration of power functions
  • Example: ∫ x^n dx, where n ≠ -1
• Integration of exponential functions
  • Example: ∫ e^x dx
• Integration of logarithmic functions
  • Example: ∫ ln(x) dx

Techniques of integration - Substitution

• Introduction to substitution method
• Basic idea: Substitute a new variable to simplify the integral
• Step-by-step process of substitution technique
• Examples with detailed solutions demonstrating substitution method
• Common substitution cases: trigonometric functions, exponents, radicals

Techniques of integration - Integration by parts

• Introduction to integration by parts
• Basic idea: Break down the integral using a product rule
• Formula for integration by parts: ∫ u dv = u v - ∫ v du
• Step-by-step process of integration by parts
• Examples with detailed solutions demonstrating integration by parts

Slide 21

• Techniques of integration - Partial fractions
  • When dealing with rational functions
  • Breaking down the fraction into simpler fractions
  • Examples:
    • ∫ (3x + 2)/(x^2 + x + 2) dx
    • ∫ (x^2 + 2)/(x^3 + x^2 - 4x) dx

Slide 22

• Techniques of integration - Trigonometric integrals
  • Basic trigonometric identities
  • Integration formulas for trigonometric functions
  • Examples:
    • ∫ sin^2(x) cos(x) dx
    • ∫ cos^3(x) dx

Slide 23

• Techniques of integration - Rationalization
  • Rationalizing expressions in the denominator
  • Examples:
    • ∫ 1/√(x+1) dx
    • ∫ 1/(x^3-1) dx

Slide 24

• Example: Integration using various techniques
  • Given function: f(x) = x^3 + 2x^2 + 5x + 7
  • Find the indefinite integral ∫ f(x) dx using different integration techniques
  • Step-by-step solution using substitution, integration by parts, and other methods

Slide 25

• Techniques of integration - Trigonometric substitutions
  • Substituting trigonometric expressions to simplify the integral
  • Common substitutions:
    • √(a^2 - x^2) → x = a sinθ
    • √(a^2 + x^2) → x = a tanθ
    • √(x^2 - a^2) → x = a secθ
  • Examples:
    • ∫ √(4 - x^2) dx
    • ∫ √(x^2 + 9) dx

Slide 26

• Techniques of integration - Improper integrals
  • Evaluating integrals with infinite limits or discontinuities
  • Examples:
    • ∫ 1/x dx, from 1 to infinity
    • ∫ 1/(x^2 + 4) dx, from 0 to infinity

Slide 27

• Techniques of integration - Numerical methods
  • Approximating the value of integrals using numerical techniques
  • Trapezoidal rule
  • Simpson’s rule
  • Example: Approximating ∫ (x^2 + 1) dx from 0 to 2 using Simpson’s rule

Slide 28

• Applications of integration - Calculating areas and volumes
  • Finding the area of irregular shapes
  • Calculating the volume of solids of revolution
  • Examples:
    • Finding the area between curves
    • Finding the volume of a cone

Slide 29

• Applications of integration - Calculating work
  • Constructing a force-distance graph
  • Calculating the work done by a varying force
  • Examples:
    • Calculating the work done by a spring
    • Calculating the work done by a variable force

Slide 30

• Recap and conclusion
  • Summary of key concepts covered in integral calculus
  • Importance of understanding and applying these concepts
  • Encouragement to practice and explore more examples and applications
  • Introduction to the next topic in the syllabus: Differential Equations