SUMMARY
The formula for finding the arclength of the curve defined by the integral of \( y = \int_{-\pi/2}^{x} \sqrt{\cos(t)} \, dt \) is derived using the arclength formula \( L = \int_{a}^{b} \sqrt{\left(\frac{dy}{dx}\right)^2 + 1} \, dx \). The derivative \( \frac{d}{dx} \int_{-\pi/2}^{x} \sqrt{\cos(t)} \, dt \) simplifies to \( \sqrt{\cos(x)} \) due to the Fundamental Theorem of Calculus, confirming that the constant limit does not affect the differentiation process. This establishes the necessary steps to compute the arclength for the specified interval.
PREREQUISITES
- Understanding of the Fundamental Theorem of Calculus
- Knowledge of arclength formulas in calculus
- Familiarity with differentiation of integrals
- Basic trigonometric functions and their properties
NEXT STEPS
- Study the application of the Fundamental Theorem of Calculus in depth
- Learn about arclength calculations for parametric curves
- Explore advanced integration techniques for non-elementary functions
- Review trigonometric identities and their derivatives
USEFUL FOR
Students studying calculus, particularly those focusing on integral applications and arclength calculations, as well as educators seeking to clarify these concepts for their students.