# Integral of a position function

• motai
In summary, the integral of a position function would physically represent the distance traveled. The first fundamental theorem of calculus is used to calculate this distance.
motai
What would the integral of a position function physically represent? I'm having a hard time trying to conceptualize this rather odd circumstance. I don't think it is used (at all), because usually problems deal with the derivatives of the position function and rates of change (velocity and acceleration) or using integration to find the initial position function (like parabolic trajectories) in the first place.

I'm wondering what would happen if we were to say integrate a parabolic trajectory (definite integral) and what answer would physically represent the outcome when the first fundamental theorem of calculus were to be applied.

I asked this question in my calculus class a while ago and didn't get a satisfactory answer.

Thanks

There are some mathematical results that are of no physical interest.

If you integrate lengths, you get an area. If you integrate areas, you get a volume.

Why wouldn't you get a length if you integrated locations/positions?

What is your variable of integration? If you integrate with respect to time you will get a quantity with units of Length*Time. I do not recognize this as having a useful physical meaning. If you set up a path integral along the trajectory you will get the distance traveled, but this is not the same as an integral wrt time.

Okay, now I see how this fits in with the Riemann Sum definition of the integral.

About the path integrals used to find distance, how is that any different from the arc length formula $$\int_a^b \sqrt{1+f'(x)^2}dx$$?

Sorry for what seems to be the silly questions... I'm just trying to push my book to the limits and questioning what the book didn't cover.

Thanks.

It isn't.That length arc formula is just a particular case of a first order curvilinear integral.

"Path integrals" is not a fortunate use of terms in this nonquantum case.

Daniel.

## 1. What is the definition of an integral of a position function?

The integral of a position function is a mathematical operation that represents the area under a curve of a position function over a given interval. It is denoted by the symbol ∫ and is used to find the displacement of an object over a specific time period.

## 2. Why is the integral of a position function important in science?

The integral of a position function is important in science because it helps to calculate the change in position of an object over time and can be used to determine important physical quantities such as velocity, acceleration, and distance traveled.

## 3. How is the integral of a position function calculated?

The integral of a position function is calculated by using a mathematical technique known as integration. This involves finding the antiderivative of the position function and then evaluating it at the upper and lower limits of the interval.

## 4. What is the relationship between the derivative and the integral of a position function?

The derivative and the integral of a position function are inversely related. This means that the derivative of a position function is equal to the velocity of the object, while the integral of the position function is equal to the displacement of the object.

## 5. Can the integral of a position function be negative?

Yes, the integral of a position function can be negative. This indicates that the object has moved in the negative direction, or in the opposite direction of the positive direction specified by the chosen coordinate system.

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