Question

Evaluate the integral: $\int_{0}^{1} |5x - 3| dx$

• A. $\frac{13}{10}$
• B. $1$
• C. $\frac{3}{10}$
• D. $\frac{1}{2}$

Hint:

Integrals of linear absolute value functions $|mx + c|$ can be evaluated geometrically without calculus! The graph forms two triangles above the $x$-axis. The total area is simply the sum of the areas of these two triangles: $\text{Area} = \frac{1}{2} \times \text{base}_1 \times \text{height}_1 + \frac{1}{2} \times \text{base}_2 \times \text{height}_2$. Here, $\frac{1}{2}(\frac{3}{5})(3) + \frac{1}{2}(\frac{2}{5})(2) = \frac{9}{10} + \frac{4}{10} = \frac{13}{10}$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem requires us to find the value of a definite integral involving an absolute value function, $|5x - 3|$, integrated over the interval from $0$ to $1$.

Step 2: Key Formula or Approach:
An absolute value function $|g(x)|$ changes sign where its argument equals zero ($g(x) = 0$). We must split the integration interval $[a, b]$ into sub-intervals at this turning point using the integral property: $$\int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx$$ The argument $5x - 3 = 0 \implies x = \frac{3}{5}$. For $x \in [0, \frac{3}{5})$, $5x - 3 < 0 \implies |5x - 3| = -(5x - 3) = 3 - 5x$. For $x \in [\frac{3}{5}, 1]$, $5x - 3 \ge 0 \implies |5x - 3| = 5x - 3$.

Step 3: Detailed Explanation:
Split the integral $I$ at $x = \frac{3}{5}$: $$I = \int_{0}^{3/5} (3 - 5x) dx + \int_{3/5}^{1} (5x - 3) dx$$ Integrate each segment independently: 1. First integral part: $$\int_{0}^{3/5} (3 - 5x) dx = \left[ 3x - \frac{5x^2}{2} \right]_{0}^{3/5}$$ Substituting the upper limit $\frac{3}{5}$: $$= 3\left(\frac{3}{5}\right) - \frac{5}{2}\left(\frac{3}{5}\right)^2 = \frac{9}{5} - \frac{5}{2}\left(\frac{9}{25}\right) = \frac{9}{5} - \frac{9}{10} = \frac{18 - 9}{10} = \frac{9}{10}$$ 2. Second integral part: $$\int_{3/5}^{1} (5x - 3) dx = \left[ \frac{5x^2}{2} - 3x \right]_{3/5}^{1}$$ Substituting limits: $$= \left( \frac{5(1)^2}{2} - 3(1) \right) - \left( \frac{5}{2}\left(\frac{3}{5}\right)^2 - 3\left(\frac{3}{5}\right) \right)$$ $$= \left( \frac{5}{2} - 3 \right) - \left( \frac{9}{10} - \frac{9}{5} \right)$$ $$= \left( -\frac{1}{2} \right) - \left( -\frac{9}{10} \right) = -\frac{1}{2} + \frac{9}{10} = \frac{-5 + 9}{10} = \frac{4}{10}$$ 3. Combine both calculated parts together: $$I = \frac{9}{10} + \frac{4}{10} = \frac{13}{10}$$

Step 4: Final Answer:
The value of the definite integral is $\frac{13}{10}$, which corresponds to option (A).