Question

$\int_{1}^{4} |x - 2| dx$ is equal to

• A. 5
• B. $\frac{7}{2}$
• C. $\frac{3}{2}$
• D. $\frac{5}{2}$

Hint:

Geometrically, the integral of \(|x-2|\) from 1 to 4 represents the area of two triangles. The first has vertices at (1,1), (2,0), (2,1) with area \(\frac{1}{2} \times 1 \times 1 = \frac{1}{2}\). The second has vertices at (2,0), (4,2), (4,0) with area \(\frac{1}{2} \times 2 \times 2 = 2\). The total area is \(\frac{1}{2} + 2 = \frac{5}{2}\).

Answer 1

The Correct Option is D

Step 1: Conceptual Foundation:
This problem involves evaluating a definite integral containing an absolute value function. The primary strategy is to partition the integral at the point where the expression within the absolute value changes its sign.
Step 2: Core Principles:
The definition of the absolute value function is provided as:
\[ |u| = \begin{cases} u & \text{if } u \ge 0
-u & \text{if } u<0 \end{cases} \]We also utilize the additive property of definite integrals: \(\int_{a}^{c} f(x)dx = \int_{a}^{b} f(x)dx + \int_{b}^{c} f(x)dx\).
Step 3: Analytical Breakdown:
The expression subject to the absolute value is \(x - 2\). This expression evaluates to zero at x = 2.


For values of \(x<2\), the term \(x - 2\) is negative, resulting in \(|x - 2| = -(x - 2) = 2 - x\).
For values of \(x \ge 2\), the term \(x - 2\) is non-negative, leading to \(|x - 2| = x - 2\).
Given that the critical point x = 2 falls within the integration interval [1, 4], we segment the integral at x = 2:
\[ \int_{1}^{4} |x - 2| dx = \int_{1}^{2} |x - 2| dx + \int_{2}^{4} |x - 2| dx \]Subsequently, we substitute the corresponding expressions for \(|x - 2|\) into each integral:
\[ = \int_{1}^{2} (2 - x) dx + \int_{2}^{4} (x - 2) dx \]The evaluation of the first integral proceeds as follows:
\[ \int_{1}^{2} (2 - x) dx = \left[ 2x - \frac{x^2}{2} \right]_{1}^{2} = \left(2(2) - \frac{2^2}{2}\right) - \left(2(1) - \frac{1^2}{2}\right) \]\[ = (4 - 2) - \left(2 - \frac{1}{2}\right) = 2 - \frac{3}{2} = \frac{1}{2} \]The evaluation of the second integral is as follows:
\[ \int_{2}^{4} (x - 2) dx = \left[ \frac{x^2}{2} - 2x \right]_{2}^{4} = \left(\frac{4^2}{2} - 2(4)\right) - \left(\frac{2^2}{2} - 2(2)\right) \]\[ = \left(\frac{16}{2} - 8\right) - \left(\frac{4}{2} - 4\right) = (8 - 8) - (2 - 4) = 0 - (-2) = 2 \]The summation of the results from the two integrals yields:
\[ \text{Total Value} = \frac{1}{2} + 2 = \frac{5}{2} \]Step 4: Concluding Result:
The definitive value of the integral is $\frac{5{2}$}.