Answer
$F\left( x \right) = \frac{{{x^3}}}{6} + 2x{\text{ }}$
Work Step by Step
$$\eqalign{
& F\left( x \right) = \int_0^x {\left( {\frac{1}{2}{t^2} + 2} \right)} dt \cr
& {\text{Integrate}} \cr
& F\left( x \right) = \left[ {\frac{1}{2}\left( {\frac{{{t^3}}}{3}} \right) + 2t} \right]_0^x = \left[ {\frac{{{t^3}}}{6} + 2t} \right]_0^x \cr
& F\left( x \right) = \left[ {\frac{{{x^3}}}{6} + 2x} \right] - \left[ {\frac{0}{6} + 0} \right] \cr
& F\left( x \right) = \frac{{{x^3}}}{6} + 2x{\text{ }}\left( {{\text{accumulation function}}} \right) \cr
& \cr
& \left( {\text{a}} \right)F\left( 0 \right) \cr
& F\left( 0 \right) = \frac{{{{\left( 0 \right)}^3}}}{6} + 2\left( 0 \right) \cr
& F\left( 0 \right) = 0,{\text{ }}\left( {{\text{Graph shown below}}} \right) \cr
& \cr
& \left( {\text{b}} \right)F\left( 4 \right) \cr
& F\left( 4 \right) = \frac{{{{\left( 4 \right)}^3}}}{6} + 2\left( 4 \right) \cr
& F\left( 4 \right) = \frac{{56}}{3},{\text{ }}\left( {{\text{Graph shown below}}} \right) \cr
& \cr
& \left( {\text{c}} \right)F\left( 6 \right) \cr
& F\left( 6 \right) = \frac{{{{\left( 6 \right)}^3}}}{6} + 2\left( 6 \right) \cr
& F\left( 6 \right) = 48,{\text{ }}\left( {{\text{Graph shown below}}} \right) \cr
& \cr
& {\text{Graphs}} \cr} $$
