Question

Use green's theorem to evaluate c f · dr. (check the orientation of the curve before applying the theorem.) f(x, y) = y − cos(y), x sin(y) , c is the circle (x − 8)2 (y 4)2 = 25 oriented clockwise

A) π
B) -π

Answer 1

The student is tasked with applying Green's Theorem to solve for an integral around a clockwise-oriented circle, where the vector field is given and the theorem's details require calculation. The student must account for the orientation when applying Green's Theorem.

Step-by-step explanation:

The student is asking to evaluate an integral using Green's Theorem, specifically, f · dr around a clockwise-oriented circle defined by (x - 8)^2 + (y - 4)^2 = 25. When using Green's Theorem, we convert a line integral around curve C into a double integral over the area D bounded by C. However, given the curve's clockwise orientation, we must account for the negative sign since Green's Theorem assumes a positive, counter-clockwise orientation.

To apply Green's Theorem, we need the partial derivatives of the vector field f(x, y) = (y - cos(y), x sin(y)). Normally, we'd find the derivatives ∂Q/∂x and ∂P/∂y and then compute the double integral ∫∫_D (∂Q/∂x - ∂P/∂y) dA. Nevertheless, without the specifics of the function components and derivatives, we can't carry out the complete evaluation here. The result, whether π or -π, depends on the correct execution of the theorem and accounting for the curve's orientation.