If - 0 . 15 0 . 15 | 100 x 2 - 1 | d x = k 3000 , then k is equal to ___________. [2023]

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Solution

Q.
If $\int_{- 0.15}^{0.15}$ $| 100 x^{2} - 1 |$ $d x = \frac{k}{3000}$ , then $k$ is equal to ___________. [2023]

Ans.
(575)

Let $I = \int_{- 0.15}^{0.15}$ $| 100 x^{2} - 1 |$ $d x$

$= 2 \int_{0}^{0.15}$ $| 100 x^{2} - 1 |$ $d x$

$(∵ \int_{- a}^{a} f (x) d x = {\begin{matrix} 2 \int_{0}^{a} f (x) d x, & if f (- x) = f (x) i.e. f (x) is even \\ 0, & if f (- x) = - f (x) i.e. f (x) is odd \end{matrix})$

$= 2 {\int_{0}^{0.1} - (100 x^{2} - 1) d x + \int_{0.1}^{0.15} (100 x^{2} - 1) d x}$

$= 2 {{[x - \frac{100 x^{3}}{3}]}_{0}^{0.1} + {[\frac{100 x^{3}}{3} - x]}_{0.1}^{0.15}}$ $= \frac{575}{3000}$

$∴$ $k = 575$

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