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AI-তৈরি নমুনা উত্তর

ধরি, $I = \int_{0}^{4} y \sqrt{4 - y}\, dy$

আমরা জানি, $\int_{0}^{a} f(y)\, dy = \int_{0}^{a} f(a - y)\, dy$
সুতরাং,
$I = \int_{0}^{4} (4 - y) \sqrt{4 - (4 - y)}\, dy$
$I = \int_{0}^{4} (4 - y) \sqrt{y}\, dy$
$I = \int_{0}^{4} \left(4 y^{1/2} - y^{3/2}\right) dy$
$I = \left[ 4 \cdot \frac{y^{3/2}}{\frac{3}{2}} - \frac{y^{5/2}}{\frac{5}{2}} \right]_{0}^{4}$
$I = \left[ \frac{8}{3} y^{3/2} - \frac{2}{5} y^{5/2} \right]_{0}^{4}$
$I = \left( \frac{8}{3} \cdot 4^{3/2} - \frac{2}{5} \cdot 4^{5/2} \right) - 0$
$I = \left( \frac{8}{3} \cdot 8 - \frac{2}{5} \cdot 32 \right)$
$I = \frac{64}{3} - \frac{64}{5} = 64 \left( \frac{1}{3} - \frac{1}{5} \right) = 64 \cdot \frac{2}{15} = \frac{128}{15}$

উত্তর: $\frac{128}{15}$ (বা $8\frac{8}{15}$)