How much work is done by a spring when its end is moved from one position x_i to another x_f? Because the force is not constant, the work equation becomes:

W = ∫ F dx, with x_i and x_f the limits on the integral. Substituting -kx for F gives:

W = - ∫ kx dx, which works out to:

W = - ½ kx² | with lower limit x_i and upper limit x_f

W = ½ kx_i² - ½ kx_f²