Key Concepts and Formulas

• Area between curves: The area bounded by $x = f(y)$, $x = g(y)$, $y = c$, and $y = d$ is given by $\int_c^d |f(y) - g(y)| \, dy$.
• Power Rule of Integration: $\int y^n \, dy = \frac{y^{n+1}}{n+1} + C$, where $n \neq -1$.
• Fundamental Theorem of Calculus: $\int_a^b f(x) \, dx = F(b) - F(a)$, where $F(x)$ is the antiderivative of $f(x)$.

Step-by-Step Solution

Step 1: Set up the integral for the area.

We are given the boundaries $y = 1$, $y = 3$, $x = 0$, and $x = y^a$. Since the boundaries are given as functions of $y$, we will integrate with respect to $y$. The area is given by the integral $\int_c^d |f(y) - g(y)| \, dy$, where $f(y)$ and $g(y)$ are the right and left boundaries, respectively, and $c$ and $d$ are the lower and upper $y$-limits. In this case, $f(y) = y^a$, $g(y) = 0$, $c = 1$, and $d = 3$. Since $y^a \geq 0$ for $y \in [1, 3]$ and $a$ being a natural number, we have $|y^a - 0| = y^a$. Therefore, the area is given by $A = \int_1^3 y^a \, dy$

Step 2: Evaluate the definite integral.

We are given that the area is $\frac{364}{3}$. So, we have $\int_1^3 y^a \, dy = \frac{364}{3}$ Using the power rule for integration, we get $\left[ \frac{y^{a+1}}{a+1} \right]_1^3 = \frac{364}{3}$ Applying the Fundamental Theorem of Calculus, we substitute the upper and lower limits: $\frac{3^{a+1}}{a+1} - \frac{1^{a+1}}{a+1} = \frac{364}{3}$ Since $1^{a+1} = 1$, we have $\frac{3^{a+1} - 1}{a+1} = \frac{364}{3}$

Step 3: Solve for a.

We need to find an odd natural number a that satisfies the equation. We can test the given options.

• If $a = 3$: $\frac{3^{3+1} - 1}{3+1} = \frac{3^4 - 1}{4} = \frac{81 - 1}{4} = \frac{80}{4} = 20 \neq \frac{364}{3}$.
• If $a = 5$: $\frac{3^{5+1} - 1}{5+1} = \frac{3^6 - 1}{6} = \frac{729 - 1}{6} = \frac{728}{6} = \frac{364}{3}$.
• If $a = 7$: $\frac{3^{7+1} - 1}{7+1} = \frac{3^8 - 1}{8} = \frac{6561 - 1}{8} = \frac{6560}{8} = 820 \neq \frac{364}{3}$.
• If $a = 9$: $\frac{3^{9+1} - 1}{9+1} = \frac{3^{10} - 1}{10} = \frac{59049 - 1}{10} = \frac{59048}{10} = 5904.8 \neq \frac{364}{3}$.

Therefore, $a = 5$ is the solution.

Common Mistakes & Tips

• Always double-check the limits of integration and the order of subtraction in the Fundamental Theorem of Calculus.
• When solving for variables in exponents, consider testing the given options, especially if the variable is constrained to be an integer.
• Be careful with arithmetic and simplification of fractions.

Summary

The problem requires finding the value of 'a' by setting up and evaluating a definite integral representing the area of a region bounded by given curves. We found that the area is given by $\int_1^3 y^a dy = \frac{364}{3}$. Evaluating the integral and testing the options, we found that $a = 5$ satisfies the given equation and the condition that a is an odd natural number.

The final answer is $\boxed{5}$, which corresponds to option (B).