A $3-$D printer lays down a semicircular arc of positively charged plastic with a radius R $=2\mathrm{.}5$ cm$,$ and a linear charge density of $\backslash$lambda $=+1\mathrm{.}1\backslash$mu C$/$m$.$ After the printer has finished the arc, the stylus moves to the center of the arc as shown. The minute segment of the plastic arc highlighted in the diagram subtends an angle d$\backslash$theta $.$ Note the measurement of the angle $\backslash$theta shown in the figure. $($a$)$Input a symbolic expression for the charge dq on the segment of charge of size d$\backslash$theta in terms of given parameters.

dq $=($b$)$Input a symbolic expression for the electric force vector, exerted by the minute segment of plastic subtending the arc d$\backslash$theta on an electron $($charge $-$e$),$ within the stylus at the center of the arc. Express your answer in terms of given parameters, fundamental constants, and the unit vectors i and j in the specified coordinate system.

dF $=($c$)$Find the indefinite integral of the x$-$component Fx from part $($b$),$ but do not evaluate the limits$.$

Fx $=($e$)$Using the limits from part $($d$)$which is $-$ pi$/2$ to pi$/2$ and the expression from part $($c$)$ to determine an expression for the magnitude of the electrical force exerted by the positively charged line of plastic on the electron in the stylus.

F $=$