Question

Use Green's Theorem to evaluate the line integral along the given positively oriented curve. C 2y3 dx − 2x3 dy C is the circle x2 + y2 = 16

Answer 1

The line integral using Green's Theorem for the given vector field around the circle is evaluated to be zero.

Step-by-step explanation:

To evaluate the line integral using Green's Theorem, we first note that the line integral is given by ∫C(2y3dx - 2x3dy). Green's Theorem relates a line integral around a simple, closed, positively oriented curve C to a double integral over the region D enclosed by C. It states that ∫C(Pdx + Qdy) = ∫∫D(∂Q/∂x - ∂P/∂y) dA, where P and Q are the components of the vector field, and D is the region enclosed by C.

In our case, P = 2y3 and Q = -2x3. Applying Green's Theorem, we find:

1. Compute ∂Q/∂x = -6x2.
2. Compute ∂P/∂y = 6y2.
3. Evaluate the double integral over the region D: ∫∫D(-6x2 - 6y2)dA.
4. Since D is the circle x2 + y2 = 16, we convert to polar coordinates with r ranging from 0 to 4 and θ from 0 to 2π.
5. The double integral simplifies to ∫∫D(-6r2)r dr dθ, which evaluates to zero.

Therefore, the value of the line integral is 0.

Answer 2

To evaluate the line integral on the given circle using Green's Theorem, parameterize the curve, find the curl of the vector field, and evaluate the double integral over the region.

Step-by-step explanation:

To evaluate the line integral along the given circle, we can use Green's Theorem. Green's Theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by the curve. The theorem states that if the components of the vector field satisfy certain conditions, then the line integral of the vector field along the curve is equal to the double integral of the curl of the vector field over the region.

In this case, the curve is the circle x^2 + y^2 = 16. To use Green's Theorem, we need to parameterize the curve and find the curl of the vector field. Let's parameterize the curve as x = 4cos(t) and y = 4sin(t), where t ranges from 0 to 2π. The curl of the vector field is ∂Q/∂x - ∂P/∂y, where P = 2y^3 and Q = -2x^3. Evaluating the double integral of the curl over the region gives us the answer.