{"id":160863,"date":"2024-11-03T13:48:00","date_gmt":"2024-11-03T13:48:00","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=160863"},"modified":"2024-11-03T13:48:03","modified_gmt":"2024-11-03T13:48:03","slug":"use-greens-theorem-to-evaluate-the-line-integral-along-the-given-positively-oriented-curve","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/11\/03\/use-greens-theorem-to-evaluate-the-line-integral-along-the-given-positively-oriented-curve\/","title":{"rendered":"Use Green’s Theorem to evaluate the line integral along the given positively oriented curve"},"content":{"rendered":"\n

Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.
\u222e
C
\u200b
(ye
x
dx+2e
x
dy)
C is the rectangle with vertices (0, 0), (3, 0), (3, 2), and (0, 2).<\/p>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

To evaluate the line integral<\/p>\n\n\n\n

[
\\oint_C \\left( y e^x \\, dx + 2 e^x \\, dy \\right)
]<\/p>\n\n\n\n

using Green’s Theorem, we first identify the functions involved and set up the theorem correctly. Green’s Theorem states that if (C) is a positively oriented, simple, closed curve in the plane and (D) is the region bounded by (C), then:<\/p>\n\n\n\n

[
\\oint_C (P \\, dx + Q \\, dy) = \\iint_D \\left( \\frac{\\partial Q}{\\partial x} – \\frac{\\partial P}{\\partial y} \\right) dA
]<\/p>\n\n\n\n

where (P = y e^x) and (Q = 2 e^x).<\/p>\n\n\n\n

Step 1: Compute the Partial Derivatives<\/h3>\n\n\n\n

We need to compute the partial derivatives of (P) and (Q):<\/p>\n\n\n\n