Allied & ICAR Exam Bank • ALLIED_1040 NAT • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Work done and line integrals

Question Statement:

The work done by a force field $\vec{F} = (2x + y)\hat{i} + x\hat{j}$ in moving a particle from $(0, 0)$ to $(2, 4)$ along the curve $y = x^2$ is:

Answer Key:

12

Step-by-Step Solution & Explanation

$W = \int_C \vec{F} \cdot d\vec{r} = \int_C (2x + y) dx + x dy$. Along $y = x^2$, $dy = 2x dx$. $W = \int_0^2 [(2x + x^2) + x(2x)] dx = \int_0^2 (2x + 3x^2) dx = \left[x^2 + x^3\right]_0^2 = 4 + 8 = 12$.
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