Confirm that \( V(x, y, z)=9 y z \cos x \) is a potential function for \( \mathbf{F}(x, y, z)=\langle-9 y z \sin x, 9 z \cos x, 9 y \cos x\rangle \). Then determine the line integral along the curve \( \mathbf{r}(t)=\left\langle\frac{\pi}{2}(t-1), \frac{\pi}{2} t, t\right\rangle, 0 \leq t \leq 1 \). Write the exact answer. Do not round.

Step 1 ：Check if \( \nabla V = \mathbf{F} \) for \( V(x, y, z) = 9yz \cos x \) and \( \mathbf{F}(x, y, z) = \langle -9yz \sin x, 9z \cos x, 9y \cos x \rangle \).

Step 2 ：Compute \( \nabla V = \langle \frac{\partial V}{\partial x}, \frac{\partial V}{\partial y}, \frac{\partial V}{\partial z} \rangle \) as \( \langle -9yz \sin x, 9z \cos x, 9y \cos x \rangle \).

Step 3 ：Find the line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \) along the curve \( \mathbf{r}(t) = \left\langle \frac{\pi}{2}(t-1), \frac{\pi}{2}t, t \right\rangle, 0 \leq t \leq 1 \) using the potential function \( V(x, y, z) = 9yz \cos x \).

Step 4 ：Compute \( V(\mathbf{r}(1)) - V(\mathbf{r}(0)) \) as \( 9\frac{\pi}{2} \cos \frac{\pi}{2} \cdot \frac{\pi}{2} - 9\frac{\pi}{2} \cos \frac{\pi}{2} \cdot 0 \).

Step 5 ：Simplify the expression to obtain the exact line integral value.