Question

derive an expression for energy stored in a stretched spring

Answer 1

Restoration Force of the spring 

(compression or tensile)
= - k x  = - external force F
 
     Work done by the external force in compressing the spring from x = 0 to x = L can be obtained by:

refer attachment...


This energy is stored as potential energy ..

Answer 2

Consider a spring that can either be compressed or stretched .

The force applied is directly proportional to the elongation
-F = kx --------eq.1

Negative sign indicates the change.

Small amount of work is done in stretching
dw = Fdx

When the force is removed object retains its original position
So , dw = -Fdx

Now using eq. 1 value of F
dw = kxdx

Now integrating it for complete work done , we get
W = 1/2kx^2