Subtopic: Physical interpretations – line, surface and volume integrals
Question Statement:
The value of the line integral $\int_C \vec{F} \cdot d\vec{r}$, where $\vec{F} = x^2\hat{i} + y^2\hat{j}$ along the straight line segment $C$ from $(0, 0)$ to $(1, 1)$, is ________ (round off to 2 decimal places).
Answer Key:
0.67
Step-by-Step Solution & Explanation
Parameterize the straight line segment joining $(0, 0)$ and $(1, 1)$:
$x = t, \quad y = t, \quad t \in [0, 1]$
Then $dx = dt$ and $dy = dt$.
Substitute into the line integral:
$\int_C \vec{F} \cdot d\vec{r} = \int_0^1 (x^2 dx + y^2 dy) = \int_0^1 (t^2 dt + t^2 dt) = \int_0^1 2t^2 dt$
$= 2 \left[\frac{t^3}{3}\right]_0^1 = \frac{2}{3} \approx 0.67$