QJEEM22Ma4370: JEE Main 2022 Mathematics - Class 12 JEEM22Ma4370 Solution: The odd natural number a, such that the area of the region…

$\mathrm{a}$ is a odd natural number and

$\left| {\int\limits_1^3 {{y^a}dy} } \right| = {{364} \over 3}$

$\Rightarrow \left| {{1 \over {a + 1}}\left( {{y^{a + 1}}} \right)_1^3} \right| = {{364} \over 3}$

$\Rightarrow {{{3^{a + 1}} - 1} \over {a + 1}} = \, \pm \,{{364} \over 3}$

Solving with $( - )$ sign,

${{{3^{a + 1}} - 1} \over {a + 1}} = {{364} \over 3} \Rightarrow (a = 5)$

Solving with $( + )$ sign,

${{{3^{a + 1}} - 1} \over {a + 1}} = {{ - 364} \over 3}$ , No a exist

$\therefore$ $(a = 5)$