Understanding Antiderivatives | The Integral of Velocity for Finding Object Displacement Over Time

Antiderivative of Velocity (Position)

The antiderivative of velocity (or position) function represents the displacement of an object over time

The antiderivative of velocity (or position) function represents the displacement of an object over time. It is also known as the integral of velocity.

To find the antiderivative of velocity, you need to integrate the velocity function with respect to time. This process involves reversing the differentiation process and finding the original function.

Let’s say the velocity function is denoted as v(t), where t represents time. To find the antiderivative, you can write it as ∫v(t) dt.

The integral of v(t) dt represents the area under the velocity curve over a certain interval of time. By integrating, you are essentially finding the total displacement of the object during that time interval.

When you integrate v(t), you will obtain a function that represents the position of the object with respect to time. This function is generally denoted as s(t), where s represents the position.

So, the antiderivative of velocity, or the position function, can be written as:

s(t) = ∫v(t) dt

It is important to note that the antiderivative of velocity may have an arbitrary constant of integration, which is denoted as C. This constant is introduced during the process of finding the antiderivative and should be included in the final solution.

Therefore, the full expression for the antiderivative of velocity is:

s(t) = ∫v(t) dt + C

To find the specific position function, you would need additional information, such as an initial condition or a definite integral over a specific interval of time.

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