Kuhan said:
[itex]s(t)[/itex] is a mathematical expression that represents the distance traveled by an object starting at time [itex]t_0[/itex] and ending at time [itex]t[/itex]. The integral symbol [itex]\int[/itex] indicates that the distance is calculated by finding the area under the curve of the function [itex]\left | R'(\xi) \right |[/itex] with respect to the variable [itex]\xi[/itex].
[itex]\xi[/itex] is a variable used to represent the different points on the curve of the function [itex]\left | R'(\xi) \right |[/itex]. It is often called the dummy variable and does not hold any specific value, but is used to indicate the independent variable in the integral.
By taking the derivative of [itex]s(t)=\int_{t_0}^{t}\left | R'(\xi) \right |d\xi[/itex] with respect to time, we can find the velocity function [itex]v(t)[/itex]. This is because the derivative of an integral is the original function, so [itex]v(t)[/itex] will be equal to [itex]\left | R'(\xi) \right |[/itex] at every point [itex]\xi[/itex] on the curve.
The absolute value ensures that the distance calculated by the integral is always positive. This is important because distance cannot be negative, and the function [itex]R'(\xi)[/itex] may have negative values at certain points on the curve, which would result in a negative distance without the absolute value.
The shape of the curve [itex]R(t)[/itex] affects the distance [itex]s(t)[/itex] in two ways. First, the distance will increase or decrease based on the magnitude of the function [itex]R'(\xi)[/itex], which represents the slope of the curve at each point [itex]\xi[/itex]. Secondly, the distance will also be affected by the length of the curve itself, as the integral is calculated over the entire length of the curve from [itex]t_0[/itex] to [itex]t[/itex]. Therefore, the shape of the curve plays a significant role in determining the total distance traveled [itex]s(t)[/itex].