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QB_DIAG_S1_004
Question Bank • QB_DIAG_S1_004
MCQ • 2 Marks
Section 1: Engineering Mathematics
Calculus
Subtopic: Vector calculus
Question Statement:
Referring to the closed triangular path $C$ with vertices $(0,0)$, $(3,0)$, and $(0,4)$ oriented counter-clockwise as shown in the vector diagram, the line integral $\oint_C (2y \, dx + 5x \, dy)$ evaluated using Green's Theorem is:
Options:
(A) 18
(B) 12
(C) 24
(D) 30
Step-by-Step Solution & Explanation
By Green's Theorem in the plane:
$\oint_C (P \, dx + Q \, dy) = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA$
Here $P(x, y) = 2y$ and $Q(x, y) = 5x$.
$\frac{\partial Q}{\partial x} = 5, \quad \frac{\partial P}{\partial y} = 2 \implies \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 5 - 2 = 3$
$\oint_C (2y \, dx + 5x \, dy) = \iint_R 3 \, dA = 3 \times \text{Area of right triangle} = 3 \times \left( \frac{1}{2} \times 3 \times 4 \right) = 3 \times 6 = 18$.
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