How to apply integrals and derivatives in physics?

Question

We have learned how to count integrals and derivatives in maths and we were told we would use this in physics but how do I know when to use integral and when I should differentiate? I know that integral is an area under the curve, but how do I know what the area represents? for eg how do i know that when i make a graph of dependence of velocity on time, that the area under the curve is the $s$ and that the derivative would be $a$?

Answer 1

You'll know from context once you've learned a bit more of the subject. One common trick is dimensional analysis. For example, do you think we get velocity by differentiating or integrating position with respect to time? Well, since $v$ has the units of $x/t$, it has the units of $dx/dt$. Similarly, $x$ has the units of $vt$, i.e. those of $\int v dt$. Oh, look! Differentiating $x$ gives $v$; integrating $v$ gives $t$. The reverse would be dimensionally inconsistent. That $a=dv/dt$ as opposed to $v=da/dt$ follows similarly.