We are given the polar equation:

$r = 2 \sin(9\theta)$

and asked to find the area enclosed by one loop of the curve.

Step 1: Formula for the area in polar coordinates

The formula for the area enclosed by a polar curve $r = f(\theta)$ over a given interval is:

$A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta$

In this case, $r = 2 \sin(9\theta)$, so we need to compute:

$A = \frac{1}{2} \int_{\alpha}^{\beta} [2 \sin(9\theta)]^2 \, d\theta$

Step 2: Simplify the expression

First, simplify $[2 \sin(9\theta)]^2$: $[2 \sin(9\theta)]^2 = 4 \sin^2(9\theta)$ Thus, the area formula becomes: $A = 2 \int_{\alpha}^{\beta} \sin^2(9\theta) \, d\theta$

Step 3: Determine the limits of integration

Since the curve is periodic, one loop of the curve corresponds to a full oscillation of the sine function. The sine function $\sin(9\theta)$ completes one cycle (one loop) when $9\theta$ goes from $0$ to $\pi$, which corresponds to $\theta$ going from $0$ to $\frac{\pi}{9}$.

Thus, the limits of integration are $\alpha = 0$ and $\beta = \frac{\pi}{9}$.

Step 4: Use a trigonometric identity

We use the identity: $\sin^2(x) = \frac{1 - \cos(2x)}{2}$ to rewrite the integral: $A = 2 \int_0^{\frac{\pi}{9}} \frac{1 - \cos(18\theta)}{2} \, d\theta$ Simplifying: $A = \int_0^{\frac{\pi}{9}} (1 - \cos(18\theta)) \, d\theta$

Step 5: Integrate

Now, integrate term by term: $\int_0^{\frac{\pi}{9}} 1 \, d\theta = \left[ \theta \right]_0^{\frac{\pi}{9}} = \frac{\pi}{9}$ $\int_0^{\frac{\pi}{9}} \cos(18\theta) \, d\theta = \frac{1}{18} \sin(18\theta) \bigg|_0^{\frac{\pi}{9}} = \frac{1}{18} \sin(\pi) - \frac{1}{18} \sin(0) = 0$

Thus, the area simplifies to: $A = \frac{\pi}{9}$

Final Answer:

The area enclosed by one loop of the curve is:

$A = \frac{\pi}{9}$

Do you want more details or have any questions?

Here are five additional related questions to deepen your understanding:

1. How would the area formula change if the function involved $\cos(9\theta)$ instead of $\sin(9\theta)$?
2. Can we calculate the area of multiple loops by adjusting the limits of integration?
3. How does the period of the sine function $\sin(n\theta)$ affect the shape and number of loops in the curve?
4. What is the geometrical interpretation of a polar curve like $r = 2 \sin(9\theta)$?
5. How does the area formula in polar coordinates compare to finding areas in Cartesian coordinates?

Tip: When calculating areas in polar coordinates, always consider symmetries in the graph to simplify limits of integration, especially for periodic functions like sine or cosine.