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Home » heavy is the head that wears the crown - Upplift

heavy is the head that wears the crown - Upplift

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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📌 = \int_0^{0.8} \int_0^{0.2} 30 s^4 ds \, du + \int_{0.8}^1 \int_0^{1 - u} 30 s^4 ds \, du

📌 First integral:

📌 \int_0^{0.2} 30 \cdot \frac{s^5}{5} ds = 6 \int_0^{0.2} s^5 ds = 6 \cdot \frac{s^6}{6} \Big|_0^{0.2} = (0.2)^6 = \left(\frac{1}{5}\right)^6 = \frac{1}{15625}

📌 Multiply by 1 (no outer u): $ \frac{1}{15625} $

📌 Second integral:

📌 \int_{0.8}^1 30 \cdot \frac{(1 - u)^5}{5} du = 6 \int_{0.8}^1 (1 - u)^5 du

📌 Let $ w = 1 - u $, $ dw = -du $, limits $ w = 0.2 $ to $ 0 $:

📌 \int_0^{0.2} w^5 dw = 6 \cdot \frac{w^6}{6} \Big|_0^{0.2} = (0.2)^6 = \frac{1}{15625}

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