Calculus Question: How do I find force from work and distance?

Work is defined as F dot d, where F is the force and d is the displacement, and dot is the dot product (don't know how to do a simple dot )

F dot d = F*d*cos(theta), where theta is the angle between the force vector and the displacement vector. Luckily, for springs, the Force is always opposite the displacement of the spring end (F = -kx), so F*d*cos(theta) is generally -F*d. This would be enough to find the force, except that the force is not constant over the distance of stretching. So you have to integrate.

The integral is of -F*dx from 0 to x. F, from Hooke's law, is -kx, thus the integral from 0 to x of kx dx is just (1/2)k*x^2. That's equal to the total work (in this case, 3 J).

From this, you can find the spring constant, k. 3 J = (1/2)*k*(13 m)^2  ==> k = 0.0355 N/m.

Now, to find the force, go back to Hooke's law: F = -kx = -(0.0355 N/m) * (13 m) = -0.462 N.

You probably just want the magnitude of the force. Hence, |F| = 0.462 N.