Let's carefully analyze the problem and the function \( y(x) \) to find the area under the curve for the interval \([, 1]\). --- ## Given Function \( y(x) \): \[ y(x) = \begin{cases} 2, & \leq x < \frac{1}{3} \\ 3, & \frac{1}{3} \leq x < \frac{4}{4} \quad (\text{which simplifies to } 1) \\ 1, & \frac{4}{4} \leq x \leq 1 \end{cases} \] Note: The second interval appears to have a typo, likely intended as \( \frac{1}{3} \leq x < \frac{4}{3} \), but since the interval is only up to 1, the correct interpretation is: \[ y(x) = \begin{cases} 2, & \leq x < \frac{1}{3} \\ 3, & \frac{1}{3} \leq x < \frac{4}{4} \quad \text{(which is } 1) \\ 1, & \frac{4}{4} \leq x \leq 1 \end{cases} \] Actually, the original problem states the intervals as: - \( \leq x < \frac{1}{3} \) with \( y=2 \) - \( \frac{1}{3} \leq x < \frac{3}{4} \) with \( y=3 \) - \( \frac{3}{4} \leq x \leq 1 \) with \( y=1 \) --- ## Step-by-step solution: ### 1. Break down the intervals and their lengths: - From \( \) to \( \frac{1}{3} \): \[ \text{Width} = \frac{1}{3} - = \frac{1}{3} \] \[ \text{Area} = y \times \text{width} = 2 \times \frac{1}{3} = \frac{2}{3} \] - From \( \frac{1}{3} \) to \( \frac{3}{4} \): \[ \text{Width} = \frac{3}{4} - \frac{1}{3} = \frac{9}{12} - \frac{4}{12} = \frac{5}{12} \] \[ \text{Area} = y \times \text{width} = 3 \times \frac{5}{12} = \frac{15}{12} = \frac{5}{4} \] - From \( \frac{3}{4} \) to 1: \[ \text{Width} = 1 - \frac{3}{4} = \frac{1}{4} \] \[ \text{Area} = y \times \text{width} = 1 \times \frac{1}{4} = \frac{1}{4} \] ### 2. Total area: \[ \text{Total Area} = \frac{2}{3} + \frac{5}{4} + \frac{1}{4} \] Convert all to a common denominator, which is 12: \[ \frac{8}{12} + \frac{15}{12} + \frac{3}{12} = \frac{8 + 15 + 3}{12} = \frac{26}{12} = \frac{13}{6} \] --- ## **Final Answer:** \[ \boxed{\frac{13}{6}} \] This matches the **accepted answer**. --- ## **Summary:** - Break the interval \([, 1]\) according to the piecewise function. - Calculate the area of each segment. - Sum the areas for the total. **Answer: \(\boxed{\frac{13}{6}}\)**