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Find a step function whose integral has specific properties

Find a step function s such that

    \[ \int_0^2 s(x) \, dx = 5, \qquad \text{and} \qquad \int_0^5 s(x) \, dx = 2. \]


Let

    \[ s(x) = \begin{cases} \phantom{-}\frac{5}{2} & \text{if } 0 \leq x < 2, \\ -1 & \text{if } -2 \leq x \leq 5. \end{cases} \]

Then,

    \begin{align*}  \int_0^2 s(x) \, dx &= 2 \cdot \frac{5}{2} = 5 \\  \int_0^5 s(x) \, dx &= 2 \cdot \frac{5}{2} + 3 \cdot -1 = 2 \end{align*}

as requested.

2 comments

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