Question

The line integral of the vector function \(u(x, y) = 2y \, \hat{i} + x \, \hat{j}\) along the straight line from (0, 0) to (2, 4) is ..........
 

Hint:

For straight-line paths, parametrize using \(x = a + (b-a)t\) and express all variables in terms of one parameter before integrating.

Answer 1

Correct Answer: 12

Step 1: Parametrize the line.
The line from \((0,0)\) to \((2,4)\) can be written as \(x = t,\ y = 2t\), where \(t\) varies from 0 to 2. Then, \(dx = dt\) and \(dy = 2dt\).

Step 2: Evaluate the line integral.
\[ \int_C \vec{u} \cdot d\vec{r} = \int_0^2 [(2y)dx + x dy] \] Substitute \(y = 2t,\ dx = dt,\ dy = 2dt,\ x = t\): \[ \int_0^2 [2(2t)(1) + t(2)] dt = \int_0^2 [4t + 2t] dt = \int_0^2 6t dt \] \[ = [3t^2]_0^2 = 12 \] Hence, the line integral = 12.

Step 3: Verify calculation.
All substitutions are correct, and path is linear; thus, final answer is 12.

Final Answer: 12