Suppose that a particle is located at the origin $(s = 0)$ at time $t = 1$ and moves along the line with velocity $v(t) = t^{-2}$.
How can I find the position s as a function of time?
And how can I show that the particle will never cross the point at $s = 2$?
Too long for a comment.
Something that I wish someone had told me when I started trying to learn calc on my own is how important it is to learn the relationships between position, velocity, and acceleration, and how the areas/tangent lines relate them all. Although these concepts don't inherently involve calculus, they help you understand the uses of the two fundamental operations of elementary calculus: finding the slope of a tangent and the area under a curve. This video is extremely instructive in conveying these ideas. Once you get that concept, you can start to realize how powerful the two major tools (the integral and the derivative) are.
Once you understand that, I would then try and go about doing this problem. However, if you just want to know calc to impress your friends/teachers/parents (been there), the integral of the velocity function gives you the position function.