The problem you have provided asks to evaluate the line integral:

$\int_C 3xy^6 \, ds$

where $C$ represents the right half of the circle defined by the equation $x^2 + y^2 = 9$.

Step-by-step solution:

1. Parameterize the Curve $C$: The equation $x^2 + y^2 = 9$ represents a circle with a radius of 3 centered at the origin. The right half of this circle corresponds to the semicircle where $x \geq 0$. We can parameterize the curve using:

   $x = 3 \cos t, \quad y = 3 \sin t$ for $t \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, as these values describe the right half of the circle.

2. Compute $ds$: The differential arc length $ds$ is given by:

   $ds = \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt$

   First, calculate the derivatives of $x(t)$ and $y(t)$:

   $\frac{dx}{dt} = -3 \sin t, \quad \frac{dy}{dt} = 3 \cos t$

   Now compute $ds$:

   $ds = \sqrt{(-3 \sin t)^2 + (3 \cos t)^2} \, dt = \sqrt{9 \sin^2 t + 9 \cos^2 t} \, dt = \sqrt{9} \, dt = 3 \, dt$

3. Substitute into the Line Integral: Now, we substitute the parameterized values of $x$, $y$, and $ds$ into the integral. The function inside the integral becomes:

   $3xy^6 = 3(3 \cos t)(3 \sin t)^6 = 3 \cdot 3 \cos t \cdot 729 \sin^6 t = 6561 \cos t \sin^6 t$

   The integral becomes:

   $\int_C 3xy^6 \, ds = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 6561 \cos t \sin^6 t \cdot 3 \, dt$

   Simplify:

   $= 19683 \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos t \sin^6 t \, dt$

4. Evaluate the Integral: The integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos t \sin^6 t \, dt$ can be solved using the symmetry of the integrand. Since $\cos t \sin^6 t$ is an odd function, the integral over the symmetric limits from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$ is 0.

Thus, the value of the line integral is:

$\boxed{0}$

If you have any questions or need further clarification, feel free to ask!

Here are some related questions that could help expand your understanding of this topic:

1. How does the symmetry of a function affect the value of an integral over symmetric limits?
2. What is the geometric interpretation of a line integral?
3. How do you parameterize curves in different coordinate systems, such as polar coordinates?
4. What is the significance of using odd and even functions in integrals?
5. How do you evaluate line integrals over closed curves?

Tip: When working with line integrals, always check the symmetry of the curve and the integrand, as it can simplify the calculation significantly.