Evaluate the following line integral

Question

∫c 2xy³ dx + 4x²y² dy along positively oriented circle with equation x² + (y + 1)² = 1

Answer 1

Use a parametric form: x = cos(θ) y = -1 + sin(θ)

dx = -sin(θ) dθ dy = cos(θ) dθ

Assuming that the integral is clockwise about the circle, [0, 2 pi)

the integral over dx turns out to be zero and the

integral of dy turns out to be 5 pi