Question

\(\int _{0}^{2}|4x-5|dx=\)

• A. \(\frac{17}{4}\)
• B. \(\frac{-17}{4}\)
• C. \(\frac{4}{17}\)
• D. \(\frac{-4}{17}\)

Hint:

Split the integral at x = 5/4 where 4x - 5 changes sign.

Answer 1

The Correct Option is A

Step 1: Geometric Method
The graph of $|4x-5|$ is a V shape with vertex at $\left(\tfrac{5}{4},0\right)$.

Step 2: Two Triangles
At $x=0$ the height is $5$, giving a triangle of area $\dfrac{1}{2}\times\dfrac{5}{4}\times5=\dfrac{25}{8}$. At $x=2$ the height is $3$, giving $\dfrac{1}{2}\times\dfrac{3}{4}\times3=\dfrac{9}{8}$.

Step 3: Total
$\dfrac{25}{8}+\dfrac{9}{8}=\dfrac{17}{4}$.

Final Answer:
The total area is 17/4. \[ \boxed{\text{(A) }\dfrac{17}{4}} \]