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We have an acceleration function and in order to find the displacement function, it would be logical to take an indefinite integral 2 times. Then we would get a function. Why is it proposed here to take a certain integral from 0 to t, because this way we will get a number that will also show the difference between the area of the graph above the X axis and the area of the graph below the axis. What's the point of that? How can a definite acceleration integral be useful in mechanics and why is an indefinite integral not used?

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  • $\begingroup$ The author has imposed boundary conditions $s(0)=0,\,v(0)=0$. The former is a translation convention; the latter is a reference frame convention. $\endgroup$
    – J.G.
    Jan 3, 2023 at 12:41

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Both indefinite and definite integrals are useful in mechanics. When you want a function then you will do an indefinite integral. When you want a number then you will either do a definite integral or you will evaluate an indefinite integral.

With indefinite integrals it is important to remember that they produce constants of integration. These constants must be determined by boundary conditions. Choosing a constant of integration for an indefinite integral is equivalent to choosing one of the bounds for a definite integral.

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