# Definite and indefinite integration in the definition of work

• etotheipi
In summary, the difference between definite and indefinite integrals is that definite integrals are used to represent a specific value of a quantity while indefinite integrals are used to represent the general form of a quantity. Work is more precisely defined as a definite integral, but the indefinite integral can be used to represent the quantity in a more general sense. In classical mechanics, the momentum can be represented by both definite and indefinite integrals, with the definite integral representing a specific value in a chosen reference frame and the indefinite integral representing the general form in any reference frame. However, in General Relativity, the concept of momentum is more complicated and requires the use of indefinite integrals.
etotheipi
This is going to sound like a silly question, but here we go anyway! I've always thought about a definite integral being used for modelling a change in some quantity whilst an indefinite integral is employed to find the defining function of that quantity.

For example, consider the force-momentum relationship $$F dt = dP$$ If we integrate definitely from t1 to t2, 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 x1 to x2 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?

etotheipi said:
This is going to sound like a silly question, but here we go anyway! I've always thought about a definite integral being used for modelling a change in some quantity whilst an indefinite integral is employed to find the defining function of that quantity.

For example, consider the force-momentum relationship $$F dt = dP$$ If we integrate definitely from t1 to t2, 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 x1 to x2 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?

In general, these things are always definite integrals.

You can, however, have a definite integral with variable bounds - usually the end point. This creates the variable KE or momentum as a force acts over a variable time interval or distance. So, for example:

##p(t) = p(0) + \int_0^t F(t')dt'##

Writing:

##p(t) = \int F(t)dt##

Is a bit sloppy, I would say.

sophiecentaur and etotheipi
PeroK said:
In general, these things are always definite integrals.

You can, however, have a definite integral with variable bounds - usually the end point. This creates the variable KE or momentum as a force acts over a variable time interval or distance. So, for example:

##p(t) = p(0) + \int_0^t F(t')dt'##

Writing:

##p(t) = \int F(t)dt##

Is a bit sloppy, I would say.

Thank you, that clears up a lot of confusion!

etotheipi said:
Is the quantity work more precisely defined as a definite integral as opposed to the indefinite definition which seems to be everywhere?
It's the value of C that seems to be the difficulty. How are you going to find it without working out, effectively, a definite integral somewhere? The indefinite integral is only half the story.

etotheipi
etotheipi said:
Thank you, that clears up a lot of confusion!
Actually, a further thought. In classical mechanics we are normally using an inertial reference frame, as defined by Newton's first law. If we pick a reference frame then the momentum has a definite value, as a function of time. That's represented by the definite integral above.

But, you might ask whether the momentum can be represented more generally, independent of any particular inertial reference frame. For that you could use the indefinite integral. Note that mathematically an indefinite integral is not a function, but an equivalence class of functions. The momentum, therefore, is also an equivalence class of functions, one for each inertial reference frame.

The quantity common to all inertial reference frames is the change in momentum. The actual momentum itself depends on choice of frame.

In this case, however, you would need to make clear that's what is meant. Once you choose a reference frame, then the integral becomes definite; and the momentum is resolved to a specific function.

In fact, not to get too far ahead, being able to think of a quantity like momentum in this way eventually becomes essential in General Relativity, where there are no global inertial reference frames. But that's a story for another day.

etotheipi

## What is definite integration in the definition of work?

Definite integration in the definition of work is a mathematical concept used to calculate the work done by a variable force over a specific interval. It involves finding the area under a curve on a graph that represents the force and displacement of an object.

## What is indefinite integration in the definition of work?

Indefinite integration in the definition of work is a mathematical concept used to find the general function that represents the force acting on an object. It is the reverse process of differentiation and is used to find the potential energy of an object.

## How is definite integration used to calculate work?

Definite integration is used to calculate work by finding the area under a force-displacement curve. This area represents the work done by the variable force on an object over a specific interval. The units of work are joules (J) and can be positive or negative depending on the direction of the force and displacement.

## What is the relationship between definite integration and the work-energy theorem?

The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy. Definite integration is used to calculate the work done by a variable force, which can then be equated to the change in kinetic energy of an object. This relationship is important in understanding the energy transfer between objects in a system.

## How is indefinite integration used in the definition of work?

Indefinite integration is used in the definition of work to find the potential energy of an object. This is because the potential energy is the anti-derivative of the force function. By finding the indefinite integral of the force function, we can determine the potential energy of an object at any given point.

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