In physics, when do you take an integral to find work? In relation to mechanics only (springs,kinetic energy)?

Answer 1

You need an integral to find the work of a force, when the latter is varying.
Then you need to sum the elementary works, given by the value of the force at some point and the elementary distance the application point has travelled, to get an accurate value of the total work in the end.
For instance, let's say we want to determine the work of the friction force on an object, which is given by k*v(velocity).
Then W = int( k * v (t) * dx) = int ( k * v(t) * v(t) * dt)
And if you know v(t), which should be a decreasing exponential towards a limit value, you can evaluate W of the varying friction force.

Answer 2

Always. Work is force*dx. True, if F is constant, that's just F(deltax), but it's an integral just the same. Even if you use a potential to find the work done---that potential is -int(Fdx), for example a spring, F=-kx, -int(-kxdx) = 1/2kx^2, or gravity F=-mg|y>, -int(-mgdy) = mgy

Answer 3

Congratulations, Sir your are a genius.