I am not familiar with surface integrals and need your help in solving this equation. How do I translate below mentioned equation in simple integrals with limits?
$$ \text{Absorption Rate: }\,\,\, K_a := ∬_A PC dA $$
Where, $P$ is permeability $(0.014 \text{cm/hr})$, $C$ is concentration $(6.34 \text{mg/L})$ and $A$ is surface area of intestine $(2\pi RL)$. I have values for $R$ $(1 \text{cm})$ and $L$ $(250 \text{cm})$, which gives an $A=1570$ cm$^2$.
Thanks so much, Krina
If both you note that $P$ the permeability is a constant and that $C$ is also a constant. So I think that
$$K_a = PC\int \int_A \mathrm dA$$
Now you can set $\mathrm dA = R\mathrm d\theta \mathrm dl$ such that
$$K_a = PC\int_{0}^{2\pi}\int_0^LR\mathrm d\theta \mathrm dl = PCR\left(\underbrace{\int_0^{2\pi}\mathrm d\theta}_{=2\pi}\right)\left(\underbrace{\int_0^L\mathrm dl}_{=L}\right) = PCA$$
This is my interpretation of what you tried to ask. If $C$ is a function of the volume or of the leght $L$ there will be an integration problem because you'll have to integrate and the integral will be an $\ln(L)$ and this is not unit free.
If you have a problem on how to construct an $\mathrm dA$ given some surface you can see This Link: Surface Integrals. See also Cilindrical Surface element and Spherical surface element. It may be very instructive for you. Think that first you vary an arc element $R\mathrm d\theta$ and then multiply by an length element $\mathrm dl$.