SUMMARY
The area of the region bounded by the polar equation r = 3 + 2sin(θ) can be calculated using the formula A = 1/2 ∫ r² dθ. The integral to evaluate is ∫ (3 + 2sin(θ))² dθ, which expands to ∫ (9 + 12sin(θ) + 4sin²(θ)) dθ. The solution involves splitting the integral into manageable parts and substituting sin²(θ) with (1 - cos(2θ))/2 to simplify the calculation. It is essential to determine the correct limits of integration to accurately represent one complete traversal around the boundary.
PREREQUISITES
- Understanding of polar coordinates and equations
- Knowledge of integral calculus, specifically integration techniques
- Familiarity with trigonometric identities, particularly sin²(θ)
- Ability to determine limits of integration for polar curves
NEXT STEPS
- Practice evaluating integrals involving polar equations
- Learn about the application of trigonometric identities in integration
- Study techniques for finding areas bounded by polar curves
- Explore advanced integral calculus topics, such as improper integrals
USEFUL FOR
Students studying calculus, particularly those focusing on polar coordinates and integration techniques, as well as educators looking for examples of area calculations in polar systems.