Let A be the area bounded by the curve y = x | x - 3 | , the x -axis and the ordinates x = - 1 and x = 2 . Then 12A is equal to _____. [2023]

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Solution

Q.
Let A be the area bounded by the curve $y = x$ $| x - 3 |$ , the $x$ -axis and the ordinates $x = - 1$ and $x = 2$ . Then 12A is equal to _____. [2023]

Ans.
(62)

$y = x$ $| x - 3 |$ $= x (3 - x), when - 1 \leq x \leq 2$

$Required area, A =$ $| \int_{- 1}^{0} (3 x - x^{2}) d x |$ $+ \int_{0}^{2} (3 x - x^{2}) d x$

$=$ $| {[\frac{3 x^{2}}{2} - \frac{x^{3}}{3}]}_{- 1}^{0} |$ $+ {[\frac{3 x^{2}}{2} - \frac{x^{3}}{3}]}_{0}^{2}$

$= (\frac{3}{2} + \frac{1}{3}) + (\frac{12}{2} - \frac{8}{3})$

$= \frac{11}{6} + \frac{10}{3}$

$Now 12 A = 12 (\frac{11}{6} + \frac{10}{3}) = 22 + 40 = 62$

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