A thin insulated wire forms a plane spiral of $N$ light turns carrying a current  $i$. The radii of inside and outside turns are equal to $a$and $\mathit{b}\mathbf{.}$ Find the magnetic moment of the spiral with a given current

Consider $dr$ width of the spiral, at a distance$r$ from the center. Number of turns in this 

The magnetic moment of current loop, having $dN$ turns is $dM = \left(dN\right) i A$

Total magnetic moment 

 $$\begin{gathered} M=\int dM \\ ={\int }_a^{b}dNiA \\ =\left(\frac{N}{b-a}\right){\int }_a^{b}dr i\times \pi r^2 \\ =\left(\frac{\pi Ni}{3\left(b-a\right)}\right)\left(b^3-a^3\right) \end{gathered}$$