Integral Set$-up$: $As$ you've probably noticed, most $of$ the integrals used $to$ compute $E-$field

from charge distributions are fairly complicated. Any realistic problem must $be$ evaluated $by$

computer. However, $no$ computer can set$-up$ the integration for you, $so$ it's important that

you $be$ able $todo$ this. For the cases $in$ this problem, set$-up$ and simplify $(butdo$ not evaluate$)$

the corresponding voltage $orE-$field integrals.

$Asan$ example, $if$ there $is$ a uniform line $of$ charge with density ${}_{L}0,0,00,0,D\{\begin{array}{cc}\text{:}\text{v}\text{e}\text{c}& (E)(vec(r))={\displaystyle \underset{0}{\overset{D}{}}}\frac{\text{o}\text{b}\text{r}\text{a}\text{c}\text{e}({}_L(vec(r{)}^{\text{'}}){)}^{{}_L}((vec(r))-\text{v}\text{e}\text{c}(r{)}^{\text{'}})obrace(dL{)}^{d{z}^{\text{'}}}}{4lon|(vec(r))-\text{v}\text{e}\text{c}(r{)}^{\text{'}}{|}^3}\end{array}$

vec$(E)(x,y,z)={\displaystyle \underset{0}{\overset{D}{}}}\frac{{}_L(x(hat(x))+y(hat(y))+[z-z^{\text{'}}](hat(z)))d{z}^{\text{'}}}{4lon(x^2+y^2+[z-z^{\text{'}}{]}^2{)}^{\frac{3}{2}}}$

$=(xhat(x)+$yhat$(y)+$zhat$(z))\frac{{}_L}{4lon}{\displaystyle \underset{0}{\overset{D}{}}}\frac{d{z}^{\text{'}}}{(x^2+y^2+[z-z^{\text{'}}{]}^2{)}^{\frac{3}{2}}}-$hat$(z)\frac{{}_L}{4lon}{\displaystyle \underset{0}{\overset{D}{}}}\frac{z^{\text{'}}d{z}^{\text{'}}}{(x^2+y^2+[z-z^{\text{'}}{]}^2{)}^{\frac{3}{2}}}$

This integral $is$ ready $tobe$ fed into the computer and evaluated.

$(a)$ A circular disk $of$ charge rests $on$ the $xy-$plane, centere