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Property 7

Area bounded ...

Question

Area bounded by the curve $y = x (x - 1)^{2}$ and x-axis is-

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1/3

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1/12

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1/2

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Solution

The correct option is A 1/12
$y = x (x - 1)^{2}$
On $x - a x i s$ $y = 0$
$x (x - 1)^{2} = 0$ when $x = 0$ and $x = 1$
$\int_{0}^{1} (x) (x - 1)^{2} = \int_{0}^{1} x^{3} - 2 x^{2} + x$
$= {[\frac{x^{4}}{4} - \frac{2 x^{3}}{3} + \frac{x^{2}}{2}]}_{0}^{1}$
$= \frac{1}{4} - \frac{2}{3} + \frac{1}{2} = \frac{3 - 8 + 6}{12}$
$= \frac{1}{12}$ sq/units

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