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GATE CSE 2022 Q2: Area under a piecewise constant function on [0, 1]

Question

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

y(x)={2if 0≤x<133if 13≤x<341if 34≤x≤1y(x) = \begin{cases} 2 & \text{if } 0 \le x < \frac{1}{3} \\ 3 & \text{if } \frac{1}{3} \le x < \frac{3}{4} \\ 1 & \text{if } \frac{3}{4} \le x \le 1 \end{cases}

Which one of the following is the area under the curve for the interval [0,1][0, 1] on the xx-axis?

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Answer: C (official GATE 2022 final answer key)

Concept used: The area under a step function is the sum of height × width over each piece.

First step: Add 2⋅13+3⋅(34−13)+1⋅142 \cdot \frac{1}{3} + 3 \cdot \left(\frac{3}{4} - \frac{1}{3}\right) + 1 \cdot \frac{1}{4}.

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Source: official GATE 2022 CS question paper and final answer key, organised by IIT Kharagpur. Question paper · Answer key

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