Substitution Rule Applied to Indefinite Integrals

When a force is applied to a linear spring, the restoring force increases proportionally with the amount of displacement. This behavior is described by Hooke’s law, which allows the work done on the spring to be determined directly from the force–displacement relationship. In this case, the force varies in a simple and predictable manner, making...

When a force is applied to a linear spring, the restoring force increases evenly with displacement. This relationship follows Hooke’s law, which enables the calculation of work done from the force–displacement curve.

For a nonlinear spring, Hooke’s law does not apply. Its restoring force changes with position, so the force–displacement gra...