SUMMARY
The integral ∫ x²√(a² - x²) dx from 0 to a can be evaluated using the substitution x = a sin(θ). This substitution transforms the integral into a more manageable form. The bounds of the integral change accordingly: when x = 0, θ = 0, and when x = a, θ = π/2. This method simplifies the evaluation of the integral significantly.
PREREQUISITES
- Understanding of trigonometric substitutions in calculus
- Familiarity with integral calculus techniques
- Knowledge of the arcsine function and its properties
- Ability to manipulate integral bounds during substitutions
NEXT STEPS
- Study the method of trigonometric substitution in integral calculus
- Learn how to change the bounds of integrals after substitution
- Practice evaluating integrals involving square roots and polynomial expressions
- Explore additional examples of integrals using the substitution x = a sin(θ)
USEFUL FOR
Students studying calculus, particularly those focusing on integral techniques and trigonometric substitutions, will benefit from this discussion.