Green's Theorem Line Integral
Criado em: 19 de dezembro de 2024
Criado em: 19 de dezembro de 2024
Problem 1: (10 points) Let and . Find the line integral of where is the simple closed curve . The line integral is taken counter-clockwise.
To evaluate the line integral where and , and is the unit circle traversed counter-clockwise, we can use Green's Theorem.
Green's Theorem states:
where is the region enclosed by .
Compute the Partial Derivatives:
Thus,
Set Up the Double Integral:
The region is the unit disk . Switching to polar coordinates where and , the integrand becomes:
The area element in polar coordinates is .
So, the integral becomes:
Evaluate the Integral:
Integrate with respect to :
Integrate with respect to :
Final Answer:
Answer:
After simplifying, the integral is five-halves times π. Thus,
∮C P dx + Q dy = (5⁄2) π