Evaluate $\int_{C} x d x+y d y+z d z$ where $C$ is the line segment from $(4,1,4)$ to $(7,-2,3)$
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Step 1 ：The given integral is a line integral over a vector field. The vector field in this case is F = . The line segment C is parameterized by the vector function r(t) = <4 + 3t, 1 - 3t, 4 - t> for 0 <= t <= 1. We can evaluate the line integral by taking the dot product of F and the derivative of r(t), and then integrating from t = 0 to t = 1.

Step 2 ：Let's parameterize the line segment C. We have t = t, x = 3*t + 4, y = 1 - 3*t, z = 4 - t. So, F = [3*t + 4, 1 - 3*t, 4 - t] and r = [3*t + 4, 1 - 3*t, 4 - t].

Step 3 ：Next, we find the derivative of r(t), which is dr = [3, -3, -1].

Step 4 ：Then, we take the dot product of F and dr, which gives us F_dot_dr = 19*t + 5.

Step 5 ：Finally, we integrate F_dot_dr from t = 0 to t = 1, which gives us the value of the line integral. The integral is \(\frac{29}{2}\).

Step 6 ：Final Answer: The value of the line integral \(\int_{C} x d x+y d y+z d z\) where C is the line segment from (4,1,4) to (7,-2,3) is \(\boxed{\frac{29}{2}}\).