Question

If \(f(x) = |5x-3|\) is defined on interval \([0,1]\), then the value of \(\int _0^1f(x)\,dx\) is...

• A. \(\frac{13}{10}\)
• B. \(\frac{9}{10}\)
• C. \(\frac{3}{10}\)
• D. \(\frac{17}{10}\)

Hint:

Split the integral at the point where 5x - 3 changes sign, x = 3/5.

Answer 1

The Correct Option is A

Step 1: Geometry
The graph of $|5x - 3|$ on $[0, 1]$ is a V shape with its vertex at $(0.6, 0)$, so the area is two triangles.

Step 2: Left triangle
At $x = 0$ the height is 3, and the base is 0.6. Area $= \frac{1}{2}\times0.6\times3 = 0.9$.

Step 3: Right triangle
At $x = 1$ the height is 2, and the base is $0.4$. Area $= \frac{1}{2}\times0.4\times2 = 0.4$.

Step 4: Total
$0.9 + 0.4 = 1.3 = \frac{13}{10}$.

Final Answer:
The integral is 13/10. This is option (A). \[ \boxed{\text{(A) }\frac{13}{10}} \]