Allied & ICAR Exam Bank • ALLIED_0059
NAT • 2 Marks
Section 1: Engineering Mathematics
Vector Calculus
Subtopic: Physical interpretations – line, surface and volume integrals
Question Statement:
The value of $\int_C (x dx + y dy)$ along any smooth curve from $(1, 1)$ to $(3, 4)$ is:
Step-by-Step Solution & Explanation
The integrand is an exact differential: $d\left(\frac{x^2 + y^2}{2}\right)$. Line integral is path independent: $\left[ \frac{x^2 + y^2}{2} \right]_{(1, 1)}^{(3, 4)} = \frac{9 + 16}{2} - \frac{1 + 1}{2} = 12.5 - 1.0 = 11.5$.
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