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Question: Find the mean value of x (a – x) over the range (0, a) –...

Find the mean value of x (a – x) over the range (0, a) –

A

a23\frac{a^{2}}{3}

B

a26\frac{a^{2}}{6}

C

a6\frac{a}{6}

D

None of these

Answer

a26\frac{a^{2}}{6}

Explanation

Solution

Mean value = ∫0ax(a−x)dx(a−0)\frac{\int_{0}^{a}{x(a - x)dx}}{(a - 0)}

= ∫0a(ax−x2)\int_{0}^{a}{(ax - x^{2})} dx

= {ax22−x33}0a\left\{ \frac{ax^{2}}{2} - \frac{x^{3}}{3} \right\}_{0}^{a}

= {a32−a33−0}=a26\left\{ \frac{a^{3}}{2} - \frac{a^{3}}{3} - 0 \right\} = \frac{a^{2}}{6}