GATE 2016 EC – Question 39
The region specified by $\{(\rho, \varphi, z) : 3 \leq \rho \leq 5, \frac{\pi}{8} \leq \varphi \leq \frac{\pi}{4}, 3 \leq z \leq 4.5\}$ in cylindrical coordinates has volume of ________
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Correct answer: 4.66 to 4.76
Explanation
In cylindrical coordinates the volume element is $\rho \, d\rho \, d\varphi \, dz$. The volume is $\int_3^5 \rho \, d\rho \times \left(\frac{\pi}{4} - \frac{\pi}{8}\right) \times (4.5 - 3) = \frac{25 - 9}{2} \times \frac{\pi}{8} \times 1.5 = 8 \times \frac{\pi}{8} \times 1.5 = 1.5\pi = 4.71$.