Answer
See explanation.
Work Step by Step
Displacements is the total area (area below $x$-axis is negative) The distance is the number of all areas where all areas are considered to be positive.
\begin{aligned}
&\text { (a) Displacement: } \int_{0}^{3}\left(t^{3}-3 t^{2}+2 t\right) d t=\frac{9}{4} \mathrm{m}\\
&\text { Distance: } \int_{0}^{3}\left|t^{3}-3 t^{2}+2 t\right| d t=\int_{0}^{1}\left(t^{3}-3 t^{2}+2 t\right) d t-\\
&\int_{1}^{2}\left(t^{3}-3 t^{2}+2 t\right) d t+\int_{2}^{3}\left(t^{3}-3 t^{2}+2 t\right) d t=\frac{11}{4} \mathrm{m}
\end{aligned}
Displacements is the total area (area below $x$-axis is negative) The distance is the number of all areas where all areas are considered to be positive.
\begin{aligned}
&\text { (b) Displacement: } \int_{0}^{3}(\sqrt{t}-2) d t=2(\sqrt{3}-3) \mathrm{m}\\
&\text { Distance: } \int_{0}^{3}|\sqrt{t}-2| d t\\
&=-\int_{0}^{3}(\sqrt{t}-2)=2(3-\sqrt{3}) \mathrm{m}
\end{aligned}
