Question

Use Green's theorem to evaluate the line integral along the given positively oriented curve.

∮ C 6sin(y)dx+6xcos(y)dy, C is the ellipse x^2+xy+y^2=9
a) 72π
b) 108π
c) 36π
d) 54π

Answer 1

The line integral along the given curve using Green's theorem is 0.

Step-by-step explanation:

To evaluate the line integral using Green's theorem, we need to find the curl of the given vector field. The vector field is F = 6sin(y)dx + 6xcos(y)dy. The curl of F is given by curl(F) = (∂Q/∂x - ∂P/∂y).

Now, let's find the partial derivatives of Q and P. Q = 6xcos(y), so ∂Q/∂x = 6cos(y). P = 6sin(y), so ∂P/∂y = 6cos(y). Plugging these values into the curl equation, we get curl(F) = 6cos(y) - 6cos(y) = 0.

Since the curl of F is 0, Green's theorem tells us that the line integral along a closed curve, like the ellipse x^2+xy+y^2=9, is equal to zero. Therefore, the value of the line integral is 0.