GATE 2022 AE – Question 31
Arc length of $x=\cos\theta,y=\sin\theta,z=\theta$ from 0 to $2\pi$ (one decimal place) is ________.
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Correct answer: 8.86 to 8.94
Explanation
**Arc length of a space curve** $\vec r(\theta)=(\cos\theta,\ \sin\theta,\ \theta)$ (a helix):
$$s=\int_0^{2\pi}\left|\frac{d\vec r}{d\theta}\right|d\theta .$$
**Derivative:**
$$\frac{d\vec r}{d\theta}=(-\sin\theta,\ \cos\theta,\ 1),\qquad\left|\frac{d\vec r}{d\theta}\right|=\sqrt{\sin^2\theta+\cos^2\theta+1}=\sqrt2 .$$
**Length:**
$$s=\int_0^{2\pi}\sqrt2\,d\theta=2\pi\sqrt2=\mathbf{8.9}.$$