Question

evaluate the line integral \ $\displaystyle \int {\partial d} p \, dx q \, dy$ \ around $\partial d$, the unit circle with positive orientation.

Answer 1

To evaluate the line integral around the unit circle, we use parametric equations to describe the circle, then substitute the differentials dx and dy with the expressions in terms of θ, and integrate from 0 to 2π.

Step-by-step explanation:

To evaluate the line integral of № p dx + q dy around the unit circle with positive orientation (ød), we should first parametrize the circle.

The unit circle can be described by the parametric equations x = cos(θ) and y = sin(θ), where θ varies from 0 to 2π for a full circle.

Then, dx = -sin(θ)dθ and dy = cos(θ)dθ. Plugging these into the line integral, we obtain the integral over θ from 0 to 2π of №(-p sin(θ) + q cos(θ))dθ.

This integral might be solved directly or by evaluating the individual components based on p and q.

The trick to these integrals is choosing the correct parametrization and understanding how the differential elements dx and dy transform under this parametrization.