Word Count : 525

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The parametric equations of the cycloid are: [ x = \alpha (\theta - \sin \theta), \quad y = \alpha (1 - \cos \theta) ] The length of a curve given parametrically by (x(\theta), y(\theta)) is: [ L = \int \sqrt{\left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2}, d\theta ] Compute the derivatives: [ \frac{dx}{d\theta} = \alpha (1 - \cos \theta), \quad \frac{dy}{d\theta} = \alpha \sin \theta ] Then: [ \left(\frac{dx}{d\theta}\right)^2 + \left(\frac{dy}{d\theta}\right)^2 = \alpha^2 (1 - \cos \theta)^2 + \alpha^2 \sin^2 \theta = \alpha^2 \big((1 - \cos \theta)^2 + \sin^2 \theta\big) ] __________ ________ ___ _______ ________ _____.
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