Question

A function \( y(x) \) is defined in the interval [0, 1] on the x-axis as
\[ y(x) = \begin{cases} 2 & \text{if } 0 \leq x<\frac{1}{3} \\ 3 & \text{if } \frac{1}{3} \leq x<\frac{3}{4} \\ 1 & \text{if } \frac{3}{4} \leq x \leq 1 \end{cases} \] Which one of the following is the area under the curve for the interval [0, 1] on the x-axis?

• A. \( \frac{5}{6} \)
• B. \( \frac{6}{5} \)
• C. \( \frac{13}{6} \)
• D. \( \frac{6}{13} \)

Hint:

When calculating the area under a piecewise function, break it into intervals and calculate the area for each segment separately.

Answer 1

The Correct Option is C

We are asked to find the area under the piecewise function \( y(x) \) in the interval [0, 1]. To do this, we will calculate the area for each segment of the function. Step 1: Area from 0 to \( \frac{1}{3} \)
For this segment, the value of the function is \( y(x) = 2 \). The area under this part of the curve is the rectangle formed by the base of length \( \frac{1}{3} \) and height \( 2 \): \[ \text{Area}_1 = 2 \times \frac{1}{3} = \frac{2}{3} \] Step 2: Area from \( \frac{1}{3} \) to \( \frac{3}{4} \)
For this segment, the value of the function is \( y(x) = 3 \). The area under this part of the curve is the rectangle formed by the base of length \( \frac{3}{4} - \frac{1}{3} = \frac{5}{12} \) and height \( 3 \): \[ \text{Area}_2 = 3 \times \frac{5}{12} = \frac{15}{12} = \frac{5}{4} \] Step 3: Area from \( \frac{3}{4} \) to 1
For this segment, the value of the function is \( y(x) = 1 \). The area under this part of the curve is the rectangle formed by the base of length \( 1 - \frac{3}{4} = \frac{1}{4} \) and height \( 1 \): \[ \text{Area}_3 = 1 \times \frac{1}{4} = \frac{1}{4} \] Step 4: Total Area
The total area under the curve is the sum of the areas from each segment: \[ \text{Total Area} = \text{Area}_1 + \text{Area}_2 + \text{Area}_3 = \frac{2}{3} + \frac{5}{4} + \frac{1}{4} \] Finding a common denominator: \[ \text{Total Area} = \frac{2}{3} + \frac{6}{4} = \frac{8}{12} + \frac{18}{12} = \frac{26}{12} = \frac{13}{6} \] Thus, the correct answer is (C).