Word Count : 445

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To find the perimeter of the cardioid \( r = a(1 - \cos \theta) \), we will use the formula for the arc length of a curve in polar coordinates: \[ L = \int_0^{2\pi} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta \] For the cardioid \( r = a(1 - \cos \theta) \), we first need to calculate \( \frac{dr}{d\theta} \). ### Step 1: Calculate \( \frac{dr}{d\theta} \) \[ r = a(1 - \cos \theta) \] Taking the derivative with respect to \( \theta \): \[ \frac{dr}{d\theta} = a \sin \theta \] ### Step 2: Set up the integral for the perimeter Substitute \( r = a(1 - \cos \theta) \) and \( ____ _____ _________ ________ ____ ________ __________ _______ ________ ___ _____.
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