Subtopic: Physical interpretations – line, surface and volume integrals
Question Statement:
For vectors $\vec{F} = 3xy\,\hat{i} - y^2\,\hat{j}$ and $\vec{r} = x\,\hat{i} + y\,\hat{j}$, the value of line integral $\int_C \vec{F} \cdot d\vec{r}$ along the parabolic curve $C: y = 2x^2$ in the $xy$-plane from $(0,0)$ to $(1,2)$ is ______
Options:
(A) -1.17
(B) 1.50
(C) -2.67
(D) 2.67
Official Answer Key:
A
Step-by-Step Solution & Explanation
On C: y = 2x², so dy = 4x dx. F·dR = Fx dx + Fy dy = 3xy dx − y² dy.
Substitute y = 2x²: 3x(2x²) dx − (2x²)²(4x dx) = 6x³ dx − 16x⁵ dx.
I = ∫₀¹ (6x³ − 16x⁵) dx = [6x⁴/4 − 16x⁶/6]₀¹ = [1.5 − 2.6667] = −1.1667 ≈ −1.17. Answer: (A) -1.17.
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