Step 1: Use areas of triangles:
The graph of $|4x-5|$ is a V with its vertex at $x = \frac54$. On $[0,2]$ we get two right triangles under the V.
Step 2: Left triangle:
Base from 0 to $\frac54$, height $|4(0)-5| = 5$. Area $= \frac12\cdot\frac54\cdot5 = \frac{25}{8}$.
Step 3: Right triangle:
Base from $\frac54$ to 2, which is $\frac34$. Height at $x=2$ is $4(2)-5 = 3$. Area $= \frac12\cdot\frac34\cdot3 = \frac98$.
Step 4: Add:
$\frac{25}{8} + \frac98 = \frac{34}{8} = \frac{17}{4}$.
Final Answer:
$\frac{17}{4}$, option (A).
\[ \boxed{\frac{17}{4} \text{ (A)}} \]