Calculus: Definite Integrals

Integration

Definite Integrals

• A definite integral is a type of integral that has a specific upper and lower bound, denoted as ∫[a, b] f(x) dx
• It represents the area between the curve of a function f(x) and the x-axis within the interval [a, b]
• The definite integral is a number, not a function
• Properties:
  • Linearity: ∫[a, b] af(x) dx = a∫[a, b] f(x) dx
  • Additivity: ∫[a, b] f(x) dx + ∫[b, c] f(x) dx = ∫[a, c] f(x) dx
  • Reversing bounds: ∫[a, b] f(x) dx = -∫[b, a] f(x) dx
• Evaluation:
  • Fundamental Theorem of Calculus (FTC): ∫[a, b] f(x) dx = F(b) - F(a), where F(x) is the antiderivative of f(x)
  • Numerical methods: Riemann sums, trapezoidal rule, Simpson's rule

Indefinite Integrals

• An indefinite integral is a type of integral that has no specific upper and lower bound, denoted as ∫ f(x) dx
• It represents the family of all antiderivatives of a function f(x)
• The indefinite integral is a function, not a number
• Properties:
  • Linearity: ∫ af(x) dx = a∫ f(x) dx
  • Sum rule: ∫ f(x) dx + ∫ g(x) dx = ∫ (f(x) + g(x)) dx
  • Chain rule: ∫ f(u) du = ∫ f(g(x)) * g'(x) dx, where u = g(x)
• Evaluation:
  • Basic integration rules:
    • Power rule: ∫ x^n dx = (x^(n+1))/(n+1) + C
    • Exponential rule: ∫ e^x dx = e^x + C
    • Trigonometric rules: ∫ sin(x) dx = -cos(x) + C, ∫ cos(x) dx = sin(x) + C
  • Substitution method: ∫ f(g(x)) * g'(x) dx = ∫ f(u) du, where u = g(x)

Definite Integrals

• Representative of the area between a function's curve and the x-axis within a specific interval [a, b]
• Denoted as ∫[a, b] f(x) dx, it is a number, not a function
• Properties include:
  • Linearity: scalable by a coefficient
  • Additivity: can be split into multiple integrals
  • Reversing bounds: changes the sign of the result
• Evaluation methods:
  • Fundamental Theorem of Calculus (FTC): relies on the antiderivative F(x)
  • Numerical methods: Riemann sums, trapezoidal rule, and Simpson's rule

Indefinite Integrals

• Represents the family of all antiderivatives of a function f(x)
• Denoted as ∫ f(x) dx, it is a function, not a number
• Properties include:
  • Linearity: scalable by a coefficient
  • Sum rule: can be combined with other integrals
  • Chain rule: allows for substitution of variables
• Evaluation methods:
  • Basic integration rules for power, exponential, and trigonometric functions
  • Substitution method: integrates composite functions by changing variables

Learn about definite integrals, their properties, and how to apply them. Understand the concept of upper and lower bounds, linearity, additivity, and reversing bounds.