Tutorial 5: Area Enclosed by Curves

Tutorial problems covering definite integrals and area calculations.

Sections

Definite Integrals (Problems 1-4)

  • Evaluating definite integrals
  • Properties of definite integrals
  • Fundamental Theorem of Calculus applications

Area Under Single Curve (Problems 5-8)

  • Area above x-axis
  • Area below x-axis (absolute value)
  • Curves crossing the x-axis

Area Between Curves (Problems 9-12)

  • Finding points of intersection
  • Setting up integrals for area between curves
  • Integrating with respect to x or y

Key Formulas

Area under curve (where $f(x) \geq 0$): $$A = \int_a^b f(x),dx$$

Area between curves (where $f(x) \geq g(x)$): $$A = \int_a^b [f(x) - g(x)],dx$$

General area: $$A = \int_a^b |f(x) - g(x)|,dx$$

Steps for Area Between Curves

  1. Find points of intersection (set $f(x) = g(x)$)
  2. Determine which function is above the other
  3. Set up the integral with proper limits
  4. Evaluate

Links