The value of 12 0 3 | x 2 - 3 x + 2 | d x is ________ . [2023]

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Solution

Q.
$The value of 12 \int_{0}^{3}$ $| x^{2} - 3 x + 2 |$ $d x is$ ________ . [2023]

Ans.
(22)

$Let I = 12 \int_{0}^{3}$ $| x^{2} - 3 x + 2 |$ $d x$

$= 12 \int_{0}^{3}$ $| (x - 2) (x - 1) |$ $d x$

$= 12 \times [\int_{0}^{1} (x^{2} - 3 x + 2) d x - \int_{1}^{2} (x^{2} - 3 x + 2) d x + \int_{2}^{3} (x^{2} - 3 x + 2) d x]$

$= 12 \times [{(\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 2 x)}_{0}^{1} + {(\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 2 x)}_{2}^{3} - {(\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 2 x)}_{1}^{2}] = 22$

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