Warmup problem for line integrals of conservative force

[JordanGo]

Homework Statement

A sleeve of mass m is constrained to move without
friction along the x-axis. The sleeve is connected to the point (0, 2) on the y-axis by a spring as shown in
the diagram below. Assume that Hooke’s “Law” is a good approximation for the restoring force exerted by
the spring, i.e. that F = −k4l, where k > 0 and 4l denotes the extension/compression of the spring from
its equilibrium (unextended) length, directed along the axis l of the spring. In this problem, assume that the
equilibrium (unextended) length of the spring is 1 unit.
Using an appropriate integration, compute the work W(x) necessary to move the mass from x = 0 to a
point x 6= 0. (Because of symmetry, you need only to consider the case x > 0.) Hint: Diagrams of forces,
projections, etc., could be very helpful here. What is the significance of the quantity W(x) in terms of energy?
(A simple answer will do.)
Note: You have computed the line integral of a nonconstant force that is not directed along the direction of
motion of an object. Later in this course, we shall extend the process to motion along curves.

Homework Equations

W=∫Fxds

The Attempt at a Solution

I know how to solve most of this question, just that I do not know what ds is. Can somebody help me?

 

[Mark44]

ds is the incremental change along the path.

The sleeve is connected to the point (0, 2) on the y-axis by a spring as shown in the diagram below.

What diagram?