The line integral of function F = yzi, in the counter-clockwise direction, along the circle x2 + y2 = 1 at z = 1 is A. \[ – 2\pi \] B. \[ – \pi \] C. \[\pi \] D. \[2\pi \]

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” option2=”\[ – \pi \]” option3=”\[\pi \]” option4=”\[2\pi \]” correct=”option3″]

The correct answer is $\boxed{\pi}$.

The line integral of a vector field $F$ over a curve $C$ is given by the formula

$$\oint_C F \cdot dr = \int_a^b F(x, y, z) \cdot (dx, dy, dz)$$

where $a$ and $b$ are the endpoints of $C$.

In this case, the vector field $F = yzi$ and the curve $C$ is the circle $x^2 + y^2 = 1$ at $z = 1$. To evaluate the line integral, we can use the parameterization

$$x = \cos t, \quad y = \sin t, \quad z = 1$$

for $0 \leq t \leq 2\pi$. Then,

$$F(x, y, z) = yzi = \sin t \cdot 1 \cdot i = \sin t i$$

and

$$dr = dx \times dy = -\sin t \, dt \times \cos t \, dt = -\sin^2 t \, dt$$

Therefore, the line integral is

$$\oint_C F \cdot dr = \int_0^{2\pi} -\sin^2 t \, dt = \int_0^{2\pi} (1 – \cos 2t) \, dt = \pi$$

The other options are incorrect because they do not take into account the direction of the curve $C$. The curve $C$ is a circle in the counter-clockwise direction, so the line integral must be positive.

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