Let's say I have this integral in cylindrical coordinates: $$\int_0^\pi 2z\rhoê_\rho d\phi$$
How is this calculated? I need some help in understanding how to interpret this integral.
- Why is there a $ê_\rho$ after $\rho$, but not a $ê_z$ after $z$? Does the $ê_\rho$ "hold" all of the $2z\rho$ as the "coefficient"?
- Can it then be written as a vector then, ie. $\int_0^\pi (2z\rho,0,0)d\phi$?
- Why is there only a $d\phi$? Since there's no $\phi$ in the function (vector?), can the $2zp$ be treated as "constants" and put outside the integral? Would the answer to the integral then be $2\pi z\rho$ or $2\pi z\rho ê_\rho$ ?
Kind of blurry questions, but the answers to them might help me understand more about coordinate basis and integrals.
The scalar factor $2z\rho$ is independent of $\phi$ and can be taken outside the integral, but the vector $\hat{e}_\rho$ is actually a function of $\phi$; at each point, it's the unit vector pointing in the $\rho$ direction at that point, so in terms of its $xyz$ components it's given by $$ \hat{e}_\rho = \begin{pmatrix} \cos\phi \\ \sin\phi \\ 0 \end{pmatrix} . $$ So what you have is the integral of a vector-valued function, and you can integrate it component-wise after writing $\hat{e}_\rho$ as above.