Question

If C is a triangle with vertices (0,0), (1,0) and (1,1) which are oriented counter clockwise, then \( \oint_C 2xydx + (x^2+2x)dy \) is equal to:

• A. \( \frac{1}{2} \)
• B. 1
• C. \( \frac{3}{2} \)
• D. 2

Hint:

For line integrals over closed loops in the plane, always consider Green's Theorem first. It's often much simpler than parameterizing each segment of the path, especially if the term \( (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) \) simplifies to a constant or a simple function.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem asks for the evaluation of a line integral over a simple closed path (a triangle). This is a perfect scenario for using Green's Theorem, which converts the line integral into a double integral over the region enclosed by the path.

Step 2: Key Formula or Approach:
Green's Theorem states: \( \oint_C P dx + Q dy = \iint_R \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA \).
From the given integral, we identify \( P \) and \( Q \): - \( P(x,y) = 2xy \) - \( Q(x,y) = x^2 + 2x \) The region \( R \) is the triangle with vertices (0,0), (1,0), and (1,1).

Step 3: Detailed Explanation:
First, we calculate the partial derivatives: \[ \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(x^2 + 2x) = 2x + 2 \] \[ \frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(2xy) = 2x \] Now, find the difference: \[ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = (2x + 2) - 2x = 2 \] Apply Green's Theorem: \[ \oint_C P dx + Q dy = \iint_R 2 \, dA = 2 \iint_R dA \] The double integral \( \iint_R dA \) represents the area of the region \(R\). The region \(R\) is a right-angled triangle with base 1 and height 1.
The area of the triangle is: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2} \] Substituting the area back into the equation: \[ \oint_C P dx + Q dy = 2 \times (\text{Area of R}) = 2 \times \frac{1}{2} = 1 \]
Step 4: Final Answer:
The value of the line integral is 1.