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A function $y(x)$ is defined in the interval $[0, 1]$ on the $x – $ axis as

$$y(x) = \left\{\begin{matrix} 2& \text{if} & 0 \leq x < \frac{1}{3} \\ 3& \text{if}& \frac{1}{3} \leq x < \frac{3}{4} & \\ 1 & \text{if} & \frac{3}{4} \leq x \leq 1& \end{matrix}\right.$$

Which one of the following is the area under the curve for the interval $[0, 1]$ on the $x – $ axis?

  1. $\frac{5}{6}$
  2. $\frac{6}{5}$
  3. $\frac{13}{6}$
  4. $\frac{6}{13}$
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Answer: Option C.

The Area of the function $y(x)$ is composed of an area of three rectangles, as shown in the above picture.

$\text{Total area} = \left[2\ast\left(\frac{1}{3} – 0 \right)\right] + \left[3\ast \left(\frac{3}{4}- \frac{1}{3} \right)\right] + \left[1\ast \left(1- \frac{3}{4} \right)\right]$

$\qquad  \qquad \quad =  \frac{2}{3} + \frac{15}{12} + \frac{1}{4} = \frac{26}{12} = \frac{13}{6}\;\text{unit}^{2}.$

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