The odd natural number a, such that the area of the region bounded by y = 1 , y = 3 , x = 0 , x = y a is 364…

Plotting the graph of the given functions, we get the shaded region as the area bounded by the curves.

Given, area of shaded region $=\frac{364}{3}$

$\Rightarrow {\int }_1^3{y}^{a}dy=\frac{364}{3}$

$\Rightarrow {\left[\frac{y^{a+1}}{a+1}\right]}_1^3=\frac{364}{3}$

$\Rightarrow \frac{3^{a+1}-1}{a+1}=\frac{364}{3}=\frac{729-1}{6}=\frac{3^6-1}{6}$

$\Rightarrow a=5$