Question

A curve is given x=a(cosθ + θsinθ ) , y=a(sinθ - θcosθ) , find the length of the arc from θ=1 to θ=α?

Answer 1

$$\dfrac{dx}{d\theta}=a(-\sin\theta+\sin\theta+\theta\cos\theta)=a\theta \cos\theta$$

$$\dfrac{dy}{d\theta}=a(\cos\theta-\cos\theta+\theta\sin\theta)=a\theta\sin\theta$$

$$\sqrt{(\dfrac{dx}{d\theta})^2+(\dfrac{dy}{d\theta})^2}=\sqrt{(a\theta \cos\theta)^2+(a\theta \cos\theta)^2}$$

$$=a\theta$$

$$l=\displaystyle\int_{1}^a\sqrt{(\dfrac{dx}{d\theta})^2+(\dfrac{dy}{d\theta})^2}d\theta$$

$$=\displaystyle\int_{1}^aa\theta d\theta=a[\dfrac{\theta^2}{2}]\begin{matrix} a \\ 1 \end{matrix}=\dfrac{a(a^2-1)}{2}$$