Allied & ICAR Exam Bank • ALLIED_1015MCQ • 2 Marks
Section 1: Engineering Mathematics
Vector Calculus
Subtopic: Green's theorem
Question Statement:
Using Green's theorem, the counter-clockwise line integral $\oint_C (xy dx + x^2 dy)$ around the boundary of the unit square $[0, 1] \times [0, 1]$ is:
Options:
(A) 0.5
(B) 1.0
(C) 1.5
(D) 2.0
Answer Key:
A
Step-by-Step Solution & Explanation
By Green's theorem, $\oint_C (P dx + Q dy) = \iint_R \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA$. Here $P = xy \implies \frac{\partial P}{\partial y} = x$, and $Q = x^2 \implies \frac{\partial Q}{\partial x} = 2x$. Thus $\iint_R (2x - x) dx dy = \int_0^1 \int_0^1 x dy dx = \int_0^1 x dx = \left[\frac{x^2}{2}\right]_0^1 = 0.5$.