Interpret the curve

The equation is given as $x = y^a$ where $a$ is an odd natural number.

The region is bounded by:

• $y = 1$
• $y = 3$
• $x = 0$
• $x = y^a$

So for each $y \in [1,3]$, $x$ varies from $0$ to $y^a$.

Set up the area integral

Area between the curves in terms of $y$ is $A = \int_{1}^{3} (x_{\text{right}} - x_{\text{left}})\,dy$ Here, $x_{\text{right}} = y^a, \qquad x_{\text{left}} = 0$

Therefore, $A = \int_{1}^{3} y^a \, dy$

Evaluate the integral

$A = \left[\frac{y^{a+1}}{a+1}\right]_{1}^{3} = \frac{3^{a+1}-1}{a+1}$

Given area is $\frac{364}{3}$

Hence, $\frac{3^{a+1}-1}{a+1} = \frac{364}{3}$

Check the given odd natural number options

Since options are $3,5,7,9$, substitute them one by one.

• For $a=3$: $A = \frac{3^4-1}{4} = \frac{81-1}{4} = \frac{80}{4} = 20$ Not equal to $\frac{364}{3}$.

• For $a=5$: $A = \frac{3^6-1}{6} = \frac{729-1}{6} = \frac{728}{6} = \frac{364}{3}$ This matches.

• For $a=7$: $A = \frac{3^8-1}{8} = \frac{6561-1}{8} = \frac{6560}{8} = 820$ Not equal.

• For $a=9$: $A = \frac{3^{10}-1}{10} = \frac{59049-1}{10} = \frac{59048}{10}$ Not equal.

Conclusion

The required odd natural number is $a=5$

So the correct option is B.