To solve the given problem, let's break down the information and solve it step-by-step.

Problem Analysis

The curve $C$ is given as the top semicircle of the equation: $x^2 + y^2 = 4$ This represents a circle of radius $2$ centered at the origin. Since we are only considering the top semicircle, the curve $C$ goes from $(2, 0)$ to $(-2, 0)$.

We are asked to evaluate the line integral: $\int_C (x + y) \, ds$ where $ds$ represents the arc length differential along the curve.

Solution

1. Parametrize the Curve $C$:

   Since we are dealing with a semicircle, we can use the parametrization: $x = 2 \cos t, \quad y = 2 \sin t$ where $t$ goes from $0$ to $\pi$ (as this range of $t$ covers the top semicircle from $(2, 0)$ to $(-2, 0)$).

2. Calculate $ds$:

   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 $\frac{dy}{dt}$: $\frac{dx}{dt} = -2 \sin t, \quad \frac{dy}{dt} = 2 \cos t$ Thus, $ds = \sqrt{(-2 \sin t)^2 + (2 \cos t)^2} \, dt = \sqrt{4 \sin^2 t + 4 \cos^2 t} \, dt = \sqrt{4 (\sin^2 t + \cos^2 t)} \, dt = 2 \, dt$

3. Express $x + y$ in Terms of $t$:

   Substitute the parametrization into $x + y$: $x + y = 2 \cos t + 2 \sin t = 2 (\cos t + \sin t)$

4. Set Up the Integral:

   Now, the line integral becomes: $\int_C (x + y) \, ds = \int_0^{\pi} 2 (\cos t + \sin t) \cdot 2 \, dt = 4 \int_0^{\pi} (\cos t + \sin t) \, dt$

5. Evaluate the Integral:

   We can split the integral: $4 \int_0^{\pi} (\cos t + \sin t) \, dt = 4 \left( \int_0^{\pi} \cos t \, dt + \int_0^{\pi} \sin t \, dt \right)$

   Calculate each integral separately:

   • $\int_0^{\pi} \cos t \, dt = [\sin t]_0^{\pi} = \sin(\pi) - \sin(0) = 0$
   • $\int_0^{\pi} \sin t \, dt = [-\cos t]_0^{\pi} = -\cos(\pi) + \cos(0) = 1 + 1 = 2$

   Therefore: $4 \left( 0 + 2 \right) = 4 \times 2 = 8$

Final Answer

$\int_C (x + y) \, ds = 8$

Would you like further details on any steps or have additional questions?

Related Questions

1. How do you parametrize a curve for a given shape?
2. What is the significance of $ds$ in line integrals?
3. How would the integral change if we considered the full circle instead of a semicircle?
4. How would we handle this if the integrand was $x - y$ instead of $x + y$?
5. How do you interpret the result of a line integral in physical terms?

Tip

When dealing with line integrals, ensure the curve parametrization is accurate, as errors here can lead to incorrect arc length differentials and integrand values.