Answer
The units for $~~\frac{da}{dx}~~$ are $~~\frac{pounds}{foot^2}$
The units for the integral $~~\int_{2}^{8}a(x)~dx~~$ are $~~pounds$
Work Step by Step
The units for $a(x)$ are $pounds/foot$ and the units for $x$ are $feet$. We can find the units for $\frac{da}{dx}$:
$\frac{pounds/foot}{foot} = \frac{pounds}{foot^2}$
The units for $~~\frac{da}{dx}~~$ are $~~\frac{pounds}{foot^2}$
When we have an integral $\int_{a}^{b}f(x)~dx$, then the units of measurement when we evaluate the integral are the product of the units for $f(x)$ and the units for $x$.
Consider the following integral:
$\int_{2}^{8}a(x)~dx$
The units for $a(x)$ are $~pounds/foot~$ and the units for $x$ are $~feet~$. Therefore, the units for the integral are $~~(pounds/foot) \times (foot)~~$ which are $~~pounds$
