To solve the integral over $C$, the upper half of the circle $x^2 + y^2 = 4$, we proceed as follows:

Step 1: Parameterize the Curve

The upper half of the circle can be parameterized using $x = 2 \cos(t)$ and $y = 2 \sin(t)$, where $t$ ranges from $0$ to $\pi$. This parameterization covers the upper half of the circle, and:

$x^2 + y^2 = (2 \cos(t))^2 + (2 \sin(t))^2 = 4$

So, the expression $x^2 + y^2$ remains constant at 4 for all $t$ in $[0, \pi]$.

Step 2: Express $ds$

The differential arc length $ds$ is given by:

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

Calculating $\frac{dx}{dt}$ and $\frac{dy}{dt}$:

$\frac{dx}{dt} = -2 \sin(t), \quad \frac{dy}{dt} = 2 \cos(t)$

$ds = \sqrt{(-2 \sin(t))^2 + (2 \cos(t))^2} \, dt = \sqrt{4 \sin^2(t) + 4 \cos^2(t)} \, dt = 2 \, dt$

Step 3: Set Up the Integral

Given that $x^2 + y^2 = 4$, we now have:

$(x^2 + y^2)^{10} = 4^{10} = 1048576$

Thus, the integral becomes:

$\int_C (x^2 + y^2)^{10} \, ds = \int_0^\pi 1048576 \cdot 2 \, dt = 2097152 \int_0^\pi dt$

$= 2097152 \cdot \pi$

Step 4: Conclusion

The value of the integral is:

$\int_C (x^2 + y^2)^{10} \, ds = 2097152\pi$

Do you have any questions or want further details on any step?

Here are 5 related questions:

1. What is the parameterization of the lower half of the circle?
2. How would the integral change if $C$ were the entire circle instead of just the upper half?
3. Can you explain how the differential arc length $ds$ is derived in more detail?
4. How would you evaluate an integral like $\int_C (x^2 + y^2)^n \, ds$ for any positive integer $n$?
5. What would happen if the curve $C$ were parameterized differently, such as using polar coordinates?

Tip: When evaluating line integrals over a circle, parameterization using trigonometric functions (sine and cosine) is often the most efficient approach.