$W=\underset{||||0}{lim}{\displaystyle \underset{i=1}{\overset{n}{}}}{W}_i$

$={\displaystyle \underset{a}{\overset{b}{}}}f(x)dx$

Now, substitute the values in the definition of work done.

$9\mathrm{.}5\sqrt[\phantom{2}]{\frac{1}{4}}={\displaystyle \underset{0}{\overset{\frac{1}{4}}{}}}F(x)dx$

$={\displaystyle \underset{0}{\overset{\frac{1}{4}}{}}}kxdx$

$=k[\frac{x^2}{2\sqrt[\phantom{2}]{2}}{]}_0^{\frac{1}{4}}$

$=\frac{k}{32\sqrt[\phantom{2}]{32}}$

$k=304$

Step $3$

Find the work done $W,$ by further compressing the spring by $\frac{1}{2}$ inch $($or $\frac{1}{24}$ foot$).$