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: Understand the concept
The absolute value changes its formula where \(5x - 3 = 0\), that is, at \(x = \frac{3}{5}\). For \(x < \frac{3}{5}\) it equals \(3 - 5x\), and for \(x > \frac{3}{5}\) it equals \(5x - 3\).

Step 2: First part
\[ \int_0^{3/5}(3 - 5x)\,dx = \left[3x - \frac{5x^2}{2}\right]_0^{3/5} = \frac{9}{5} - \frac{9}{10} = \frac{9}{10} \]

Step 3: Second part
\[ \int_{3/5}^{1}(5x - 3)\,dx = \left[\frac{5x^2}{2} - 3x\right]_{3/5}^{1} = \left(\frac{5}{2} - 3\right) - \left(\frac{9}{10} - \frac{9}{5}\right) = -\frac{1}{2} + \frac{9}{10} = \frac{2}{5} \]

Step 4: Add
\(\frac{9}{10} + \frac{4}{10} = \frac{13}{10}\), option (A). The value \(\frac{9}{10}\) is only the first part, and \(\frac{17}{10}\) would come from a sign error in the second part.

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