Math Problem Statement
Solution
To evaluate the line integral using Green's Theorem, let's break it into steps.
Step 1: Green's Theorem
Green's Theorem states: where:
- is the positively oriented boundary of region ,
- ,
- .
Step 2: Compute Partial Derivatives
We compute:
- ,
- .
Thus:
Step 3: Set Up the Double Integral
The region is the square with vertices . For this square:
- ,
- .
The double integral becomes:
Step 4: Evaluate the Double Integral
First, integrate with respect to :
Now, integrate with respect to :
Final Answer:
The value of the line integral is:
Would you like additional details or an explanation for any part of this? Here are related questions to consider:
- What does Green's Theorem imply geometrically?
- How does the orientation of the curve affect the result?
- Could this problem be evaluated without Green's Theorem? How?
- How would the result change if the region were a triangle instead of a square?
- What role do partial derivatives play in Green's Theorem?
Tip: Always confirm the orientation (positive or negative) of the curve when applying Green’s Theorem!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Line Integrals
Double Integrals
Partial Derivatives
Formulas
Green's Theorem: ∫C (P dx + Q dy) = ∬R (∂Q/∂x - ∂P/∂y) dA
Theorems
Green's Theorem
Suitable Grade Level
Undergraduate Calculus
Related Recommendation
Green's Theorem: Line Integral Evaluation for \( \oint_C y^2 dx + xy dy \)
Evaluate Line Integral Using Green's Theorem for a Triangle
Evaluating Line Integrals Using Green's Theorem for the Region Bounded by y=x and y=x^2
Applying Green's Theorem to Evaluate a Line Integral with Circle C
Green's Theorem: Evaluate Line Integral for Triangle Region