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Here is a breakdown of the mathematical problems from the image you uploaded:

7. **Determine the length of the loop of the curve** \( r = a(\theta^2 - 1) \).
   
8. **Show that the arc of the upper half of the curve** \( r = a(1 - \cos\theta)^n \) **is bisected at** \( \theta = \frac{2\pi}{3} \).

9. **Determine the perimeter of the curve** \( r = a \sin^3\left(\frac{\theta}{3}\right) \), where \( 0 < \theta < 3\pi \).

10. **Find the length of the spiral** \( r = e^{\alpha \theta} \) **from the pole to the point** \( (r, \theta) \).

Would you like help with a specific problem from this set? Feel free to let me know which one you'd like to focus on first, or if you'd like detailed explanations for all!

Determine the length of the loop of the curve r = a(θ² - 1), Show that the arc of the upper half of the curve r = a(1 - Cosθ)^n is bisected at θ = 2π/3, Determine the perimeter of the curve r = a sin³(θ/3), Find the length of the spiral r = e^αθ from the pole to the point (r, θ).

Explore advanced calculus problems involving polar curves, including determining the length of loops, perimeter of curves, and the bisected arc length at specific angles. Suitable for university students studying polar coordinates and curve analysis.

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