- #1

etotheipi

For example, consider the force-momentum relationship $$F dt = dP$$ If we integrate definitely from t

_{1}to t

_{2}, we will get a change in momentum or impulse:$$\int_{t_{1}}^{t_{2}} F dt = \int_{t_{1}}^{t_{2}} dP = P_{2} - P_{1} = \Delta P = I$$whilst a definite integral will instead allow us to pick a constant of integration to find the function which outputs the momentum at any given time:$$\int F dt = P + C$$It is a commonly used definition that work is the integral of force with respect to displacement, but it is usually written as an indefinite integral. This is confusing to me, since the indefinite integral of a force with respect to displacement appears to define the kinetic energy of the body:$$\int F dx = \int mv dv = E_{k} + C$$

On the other hand, a definite integral from position x

_{1}to x

_{2}defines a

*change*in kinetic energy, and the net work is also commonly defined as the change in kinetic energy of a body, like the following

$$W = \int_{x_{1}}^{x_{2}} F dx = E_{k2} - E_{k1} = \Delta E_{k}$$

Is the quantity work more precisely defined as a definite integral as opposed to the indefinite definition which seems to be everywhere?