- #31
Justabeginner
- 309
- 1
cos(pi/2)= 0
cos(-pi/2)= 0
tan(pi/4)=1
tan(-pi/4)= -1 ?
cos(-pi/2)= 0
tan(pi/4)=1
tan(-pi/4)= -1 ?
Length of a Curve: Solve Homework Equation
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SUMMARY
The discussion focuses on calculating the length of a curve defined by the integral equation \( y = \int_{-\pi/2}^x \sqrt{\cos t} \, dt \) for \( x \) in the range \([- \pi/2, \pi/2]\). Participants clarify the application of the arc length formula, specifically \( ds = \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \), and the fundamental theorem of calculus, which states that \( F'(x) = f(x) \) for \( F(x) = \int_a^x f(t) \, dt \). The correct derivative \( \frac{dy}{dx} = \sqrt{\cos x} \) is established, leading to the integration of the arc length formula to find the length of the curve.
PREREQUISITES- Understanding of integral calculus, specifically definite integrals.
- Familiarity with the fundamental theorem of calculus.
- Knowledge of arc length formulas in calculus.
- Basic trigonometric identities and their derivatives.
- Study the fundamental theorem of calculus in detail.
- Learn how to derive and apply the arc length formula for different functions.
- Practice integration techniques for trigonometric functions, particularly involving square roots.
- Explore the properties of definite integrals and their applications in curve length calculations.
Students studying calculus, particularly those focused on integral calculus and applications in geometry, as well as educators looking for examples of curve length calculations.
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