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Home » Stuck in a Labyrinth of Desires: The Untold Story of the Harem Betrayal! - Sterling Industries

Stuck in a Labyrinth of Desires: The Untold Story of the Harem Betrayal! - Sterling Industries

📅 April 21, 2026 👤 scraface

Apr 21, 2026

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📌 Question:** An environmental scientist models soil compaction using the vector field \( \vec{F}(x, y) = \langle -y, x \rangle \), representing rotational stress. Find the work done by this field along the circular path \( \vec{r}(t) = \langle 3\cos t, 3\sin t \rangle \) for \( t \in [0, 2\pi] \).

📌 The work is given by the line integral:

📌 W = \int_C \vec{F} \cdot d\vec{r}

📌 Let \( \vec{r}(t) = \langle 3\cos t, 3\sin t \rangle \), so:

📌 \vec{r}'(t) = \langle -3\sin t, 3\cos t \rangle, \quad t \in [0, 2\pi]

📌 Now, evaluate \( \vec{F}(\vec{r}(t)) = \langle -3\sin t, 3\cos t \rangle \)

📌 \vec{F} \cdot \vec{r}' = (-3\sin t)(-3\sin t) + (3\cos t)(3\cos t) = 9\sin^2 t + 9\cos^2 t = 9(\sin^2 t + \cos^2 t) = 9

📌 Thus, the integral becomes:

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