I'm studying kinematics with varying acceleration. To calculate the displacement from the acceleration you have to integrate the acceleration with respect to t, then integrate that with respect to t, this time with limits.
I've been writing this:
But it looks a little messy. Is there a better way?
The notation on this webpage is good but seems to be aimed at having a) limits on both integrals (for me the inner integral is indefinite) and b) different variables – in differentiating with respect to t both times.
Best Answer
No, there is no better notation - the double-integral notation is standard. However, the way you've written it is problematic. Notice that when you do an indefinite integral, you get a $+c$ at the end. This is a constant, so when integrated again we have $+ct$. Evaluating from $2$ to $5$, this gives a $+3c$ at the end of your answer - which you really don't want, since your answer should be a number.
In a double integral, the inner integral should a) always be definite and b) be with respect to a different variable than the outer integral. In your case, recall that velocity is not the indefinite integral of acceleration - it's $v_0 + \int_0^ta(s)ds$, where $t$ is the time. So what you want is $\int_2^5\int_0^ts^2dsdt$.
This distinction between $s$ and $t$ is important - without it, you'll run into ambiguities as to which $t$ each $dt$ applies to.