Question

The vector function \( F(r) = -x \hat{i} + y \hat{j} \)  is defined over a circular arc  \( C \)  shown in the figure. The line integral of \( \int_C F(r) \, dr \) is: 

• A. \( \frac{1}{2} \)
• B. \( \frac{1}{4} \)
• C. \( \frac{1}{6} \)
• D. \( \frac{1}{3} \)

Hint:

To evaluate line integrals, parametrize the curve, and express the vector function in terms of the parametric variables. Then, perform the integration over the given limits.

Answer 1

The Correct Option is A

We are given the vector function: \[ F(r) = -x \hat{i} + y \hat{j}, \] which is defined over a circular arc \( C \) with the angle \( 45^\circ \) and radius 1. We are asked to evaluate the line integral of \( F(r) \) along the curve \( C \). Step 1: Parametrize the circular arc.
The vector function \( F(r) \) is expressed in Cartesian coordinates. The arc \( C \) corresponds to a segment of a circle with radius 1, so we can parametrize the curve in terms of the angle \( \theta \) (where \( 0 \leq \theta \leq 45^\circ \)): \[ x = \cos(\theta), \quad y = \sin(\theta). \] Step 2: Write the line integral.
The line integral is given by: \[ \int_C F(r) \, dr = \int_C (-x \hat{i} + y \hat{j}) \cdot (dx \hat{i} + dy \hat{j}). \] Since \( x = \cos(\theta) \) and \( y = \sin(\theta) \), we substitute these into the integral: \[ \int_0^{\frac{\pi}{4}} \left( -\cos(\theta) \, d(\cos(\theta)) + \sin(\theta) \, d(\sin(\theta)) \right). \] Step 3: Solve the integral.
After solving the integral, we find that the value of the line integral is \( \frac{1}{2} \). Final Answer: \[ \boxed{\frac{1}{2}}. \]