Last time I saw an integral was something like 10 years ago, and I am having doubts about the notation I should use. I want to describe the evolution of the volume difference between two cylinders over time (representing the elastic recoil of the aorta). Below is a graphical representation:
So taking $R$ as the radius at time $t_0$, and $r$ as the radius at time $t_1$, I can describe the difference between the cylinders (which is in fact equal to flow) defined from those radii as follows:
$$\dot{Q}_\text{aorta} = \pi hR^2 - \pi hr^2$$
Now, my problem is just that I would like to express that as a definite integral over time instead of the $F(b)-F(a)$ form. I now that the function will be $2\pi hr$.
but, $\displaystyle\dot{Q}_\text{aorta}=\int_{t_0}^{t_1}\cdots$?
My problem especially lies as in how to write the variable: $\dot{r}$? $r(t)$?
And how about the $d$ part: $dt$? $dr(t)$? $\dfrac{dr}{dt}$?
I am sorry if this seems stupid, but I was not able to find some definite info.
For ease of understanding; let's assume $r(t) = at^2 + bt + c$ to be the radial change wrt time.
You'll get the integral to be:
$$\begin{align} I &= \int_{r_1}^{r_2}{2 \pi \cdot r \cdot h \cdot dr} \\ &= 2\pi h \int_{t_1}^{t_2}{(at^2 + bt + c)\left[(2at + b)dt\right]} \end{align}$$
In other words:
$$ \dfrac{dr}{dt} = 2at + b \\ \equiv dr = (2at + b) dt$$