It seems OK except for one small mistake. I think it should be $\displaystyle \frac{9}{2} \left[ t {\color{red}-} \frac{\sin (2t)}{2} \right]$, NOT $\displaystyle \frac{9}{2} \left[ t {\color{red}+} \frac{\sin (2t)}{2} \right]$. You will need to fix the other mistakes that follow from this. The final numerical answer is unchanged.
Since this is a DEFINITE Integral you should change bounds as you change the variables. There's no need to draw that triangle and figure out the $\displaystyle \sin (2t)$. The t-bounds are trivial, use them instead and it will save you a lot of work and time.