$($a$)$ Write the first three nonzero terms for the Maclaurin series of $sin(2x).$

$[5$ points$]$

$($b$)$ Use the result of part $($a$)$ to obtain an approximate answer for

$I={\displaystyle \underset{0}{\overset{\frac{}{2}}{}}}sin(2x)dx$

$[3$ points$]$

$($c$)$ Now compute the exact integral and determine how well you did by computing the relative difference, ${}_{\text{r}\text{e}\text{l}},$ between I and your approximation via

${}_{\text{r}\text{e}\text{l}}=|\frac{I-\text{a}\text{p}\text{p}\text{r}\text{o}\text{x}\text{i}\text{m}\text{a}\text{t}\text{i}\text{o}\text{n}}{\text{I}}|$

$[2$ points$]$