Question

$$\int_{0}^{2} |2x - 3| \, dx =$$

• A. $\frac{3}{10}$
• B. $\frac{5}{2}$
• C. $\frac{10}{3}$
• D. $\frac{2}{5}$

Hint:

Definite integrals of linear modulus functions like $|Ax + B|$ can be interpreted geometrically as calculating the area of two triangles! Sketching the line reveals two right-angled triangles with bases $\frac{3}{2}$ and $\frac{1}{2}$, and heights $3$ and $1$. Calculating their areas gives: $\frac{1}{2}\left(\frac{3}{2}\right)(3) + \frac{1}{2}\left(\frac{1}{2}\right)(1) = \frac{9}{4} + \frac{1}{4} = \frac{5}{2}$. This avoids integration completely!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are asked to evaluate the definite integral of an absolute value function $\int_{0}^{2} |2x - 3| \, dx$. Because of the modulus operator, the behavior of the integrand function changes depending on where the term inside the absolute value brackets changes sign.

Step 2: Key Formula or Approach:
Find the critical turning point by setting the inner expression to zero: $$2x - 3 = 0 \implies x = \frac{3}{2}$$ Since $x = \frac{3}{2}$ lies inside our integration interval $[0, 2]$, we split the integral using the standard property: $$\int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx$$ For $x \in \left[0, \frac{3}{2}\right]$, $2x - 3 \le 0 \implies |2x - 3| = -(2x - 3) = 3 - 2x$. For $x \in \left[\frac{3}{2}, 2\right]$, $2x - 3 \ge 0 \implies |2x - 3| = 2x - 3$.

Step 3: Detailed Explanation:
Let's write out the split definite integrals: $$I = \int_{0}^{\frac{3}{2}} (3 - 2x) \, dx + \int_{\frac{3}{2}}^{2} (2x - 3) \, dx$$ Integrate each section independently using the standard power rules: $$I = \left[ 3x - x^2 \right]_{0}^{\frac{3}{2}} + \left[ x^2 - 3x \right]_{\frac{3}{2}}^{2}$$ Evaluate the first integral term at its boundaries: $$\left[ 3\left(\frac{3}{2}\right) - \left(\frac{3}{2}\right)^2 \right] - [0] = \frac{9}{2} - \frac{9}{4} = \frac{18 - 9}{4} = \frac{9}{4}$$ Evaluate the second integral term at its boundaries: $$\left[ (2^2 - 3(2)) \right] - \left[ \left(\frac{3}{2}\right)^2 - 3\left(\frac{3}{2}\right) \right] = (4 - 6) - \left( \frac{9}{4} - \frac{9}{2} \right) = -2 - \left( -\frac{9}{4} \right) = -2 + \frac{9}{4} = \frac{1}{4}$$ Sum the two evaluated components to find the total value: $$I = \frac{9}{4} + \frac{1}{4} = \frac{10}{4} = \frac{5}{2}$$

Step 4: Final Answer:
The value of the definite integral is $\frac{5}{2}$, which corresponds to option (B).