Evaluate \(\int_{3}^{2} \int_{0}^{\frac{\pi}{2}} \enspace (y + y^{2}cosx ) \enspace dx dy\)

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Solution:

\( = \int_{3}^{2} [xy + y^{2}sinx ]_{0}^{\frac{\pi}{2}} \enspace dy\)

\( = \int_{3}^{2} (\frac{\pi}{2}y + y^2sin(\frac{\pi}{2})) \enspace dy\)

\( = \int_{3}^{2} (\frac{\pi}{2}y + y^2) \enspace dy\)

\( = [\frac{\pi}{4}y^2 + \frac{y^3}{3}]_{3}^{2}\)

\( = \frac{\pi}{4} . 4 + \frac{8}{3} – (\frac{\pi}{4} . 9 + \frac{27}{3})\)

\( = -\frac{5\pi}{4}-\frac{19}{3}\)

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