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Find an interval of integration to make some integral equations true

by RoRi

Find all real numbers c such that the following equations hold:

  1. \displaystyle{\int_0^c x(1-x) \, dx = 0}.
  2. \displaystyle{\int_0^c \left| x(1-x) \right| \, dx = 0}.
  1. We evaluate the integral:

        \[ \int_0^c x(1-x) \, dx = 0 \ \implies \ \left. \left( \frac{-x^3}{3} + \frac{x^2}{2} \right) \right|_0^c = 0 \implies \ \frac{-c^3}{3} + \frac{c^2}{2} = 0. \]

    Solving for c we have,

        \[ 3c^2 - 2c^3 = 0 \quad \implies \quad c = 0, \text{ or } c = \frac{3}{2}. \]

  2. Here, since \left| x(1-x) \right| \geq 0 for all x, we can use the comparison theorem (Theorem 1.20 in Apostol) to see that if \left| x(1-x) \right| > 0 for any x then,

        \[ \int_0^c \left| x(1-x) \right| \, dx > \int_0^c 0 \, dx = 0. \]

    So, for the equation to hold we must have

        \[ \left|x(1-x) \right| = 0 \quad \text{for all } 0 \leq x \leq c. \]

    Since the expression will be nonzero for any 0 < x < 1, we must have c = 0.

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