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The integral ∮_C (y dx − x dy) is evaluated along the circle x² + y² = 1/4 traversed in counter clockwise direction. The integral is equal to
- 0
- −π/4
- −π/2
- π/4
Correct answer: −π/2
Solution
The integral represents the area enclosed by the curve, and since the curve is a circle of radius 1/2, the area is π times the square of the radius, which gives us π/4. However, the orientation of the curve and the specific form of the integral leads to a negative sign, resulting in the final value of -π/2.
Related GATE Engineering Mathematics questions
- A vector field p and a scalar field r are given by
p = (2x² - 3xy + z²) î + (2y² - 3yz + x²) ĵ + (2z² - 3xz + x²) k̂
r = 6x² + 4y² - z² - 9xyz - 2xy + 3xz - yz
Consider the statements P and Q.
P: Curl of the gradient of the scalar field r is a null vector.
Q: Divergence of curl of the vector field p is zero.
Which one of the following options is CORRECT?
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