Answer
$6.75$ ft-lb is required
Work Step by Step
We are given the work required to stretch one foot beyond the natural length of the spring ($12$ft-lb), before we can answer the question we need to find the spring constant
$12=\int_{0}^{1}(kx)dx=k[\frac{x^{2}}{2}]_{0}^{1}=\frac{k}{2}=12$
$k=24$
Now that we know the spring constant, we can now find the work required to stretch 9 inches ($\frac{3}{4}ft$) beyond natural length
$W=\int_{0}^{.75}(24x)dx=6.75$ ft-lb
