Evaluate $\int_{C}\left(x^{2}+y^{2}\right) d s, C$ is the top half of the circle with radius 3 centered at $(0,0)$ and is traversed in the clockwise direction.
Solution
Step 1 ：The integral is a line integral over a scalar field. The scalar field is \(f(x, y) = x^2 + y^2\) and the path \(C\) is the top half of the circle with radius 3 centered at the origin. The line integral of a scalar field over a path \(C\) is defined as \(\int_C f ds\), where \(ds\) is the differential arc length along the path.
Step 2 ：In this case, the path \(C\) can be parameterized by \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\), where \(r = 3\) is the radius of the circle and \(\theta\) ranges from 0 to \(\pi\) because we are only considering the top half of the circle. The differential arc length \(ds\) can be expressed in terms of \(d\theta\) as \(ds = r d\theta\).
Step 3 ：Substituting these into the integral, we get \(\int_{0}^{\pi} f(r\cos(\theta), r\sin(\theta)) r d\theta\).
Step 4 ：Substituting \(r = 3\) and \(f = 9\sin(\theta)^2 + 9\cos(\theta)^2\) into the integral, we get \(\int_{0}^{\pi} 27 d\theta\).
Step 5 ：The integral evaluates to \(27\pi\). This is the final answer.
Step 6 ：Final Answer: \(\boxed{27\pi}\)
