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I've been working on an exercise where I'm asked to calculate the volume of a cylinder using cylindrical coordinates. I've already got the correct result according to the book, but I don't understand the limits that I used to get the correct answer.

The exersise is: Find the volume under $z=3x$, above $z=0$ and inside $x^2+y^2=25$

a busy cat

I don't understand why I need to integrate between $-\frac{\pi}{2} and \frac{\pi}{2}$ because if its a circumference I would need to go from 0 to $2\pi$ to have the entire circumference and then integrate from there all the height of the cylinder. Using $-\frac{\pi}{2} to \frac{\pi}{2}$ feels like I'm only calculating half of the circumference

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This is because of the additional constraint you introduce: \begin{cases} z = 3r\cos{\theta} \\ z\ge0 & \end{cases}

This implies $\cos{\theta} \ge 0$ and thus $-\frac{\pi}{2} \le \theta \le \frac{\pi}{2}$.

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  • $\begingroup$ Thanks, I feel stupid now not seeing that it was this simple haha $\endgroup$ – Roger Vallès Oct 6 '17 at 8:43

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