2.) Let C be the upper half of the circle x^2 + y^2 = 4 in R2
Compute the integral C (x^2+y^2)^10 ds.


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.

Let C be the upper half of the circle x^2 + y^2 = 4 in R2. Compute the integral C (x^2+y^2)^10 ds.

Learn how to compute the line integral of (x^2 + y^2)^10 over the upper half of the circle x^2 + y^2 = 4. This solution covers parametric equations, arc length formula, and line integrals. Suitable for college-level students studying calculus.

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