Answer
$$\left( {\text{a}} \right)d = 2{\text{ft}},{\text{ }}\left( {\text{b}} \right)T = 2{\text{ft}}$$
Work Step by Step
$$\eqalign{
& {\text{Let the velocity }}v\left( t \right) = \frac{1}{{\sqrt t }},{\text{ in feet per second, }} \cr
& {\text{for 1}} \leqslant t \leqslant 4 \cr
& \left( {\text{a}} \right){\text{The displacement in feet is given by:}} \cr
& d = \int_a^b {f\left( t \right)} dt \cr
& d = \int_1^4 {\frac{1}{{\sqrt t }}} dt \cr
& d = \int_1^4 {{t^{ - 1/2}}} dt \cr
& d = \left[ {2\sqrt t } \right]_1^4 \cr
& d = 2\sqrt 4 - 2\sqrt 1 \cr
& {\text{simplifying}} \cr
& d = 4 - 2 \cr
& d = 2{\text{ft}} \cr
& \cr
& \left( {\text{b}} \right){\text{ The total distance traveled is given by:}} \cr
& T = \int_a^b {\left| {f\left( t \right)} \right|} dt \cr
& T = \int_1^4 {\left| {\frac{1}{{\sqrt t }}} \right|} dt \cr
& \frac{1}{{\sqrt t }} > 0{\text{ for all real number }}t \cr
& {\text{then}} \cr
& T = \int_1^4 {\frac{1}{{\sqrt t }}} dt \cr
& T = 2{\text{ft}} \cr} $$
