How to calculate cylinder volume using cylinder equation $x^2+y^2 \le r^2$?
I know it's a triple integral and I think the function to integrate will be just $1$, but I have trouble figuring out the bounds as always. Is it enough to merely solve for each variable and see what their min and max values are on the domains of the parameters?
You would need to define the height of the cylinder as well. Here your equation only describe a circle.
A typical cylinder would be defined by : $$ \{ (x,y,z) \in \mathbb{R}^3\ |\ x^2 + y^2 \leq R^2, \ 0 \leq z \leq Z \} $$ For some $R,Z \geq 0 $ constant where $R$ controls the radius of the cylinder and $Z$ its height. (Here I assume the base lies in the plane $z=0$)
In which case using cylindrical coordinates $(r,\theta,z)$ would give : $$ \int_0^{2\pi} d\theta \int_0^R dr \int_0^Z dz\ 1 \cdot r $$ Where we integrate $1$ to get the volume and $r$ is the Jacobian of the change of coordinate.