Answer
See explanation.
Work Step by Step
Displacements is the total area (area below $x$-axis is negative) The gap is the number of all areas where all areas are located.
Assumed to be positive.
\begin{aligned}
&\text { (a) Displacement: } \int_{0}^{4}(t-\sqrt{t}) d t=\frac{8}{3} \mathrm{m}\\
&\text { Distance: } \int_{0}^{4}|t-\sqrt{t}| d t=-\int_{0}^{1}(t-\sqrt{t}) d t+\int_{1}^{4}(t-\\
&\sqrt{t}) d t=3 \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} \frac{1}{\sqrt{t+1}} d t=\left.(2 \sqrt{t+1})\right|_{0} ^{3}\\
&=4-2=2 \mathrm{m}\\
&\text { Distance: } \int_{0}^{3} \frac{1}{\sqrt{t+1}} d t=\left.(2 \sqrt{1+t})\right|_{0} ^{3}\\
&=-2+4=2 \mathrm{m}
\end{aligned}
