4. (12 points) Use the Fundamental Theorem of Line Integrals to evaluate \[ \int_{C} \nabla f \cdot d \mathbf{r} \] where $f(x, y, z)=e^{x y z}+\arccos (x+y)-z^{2}$, and $\mathbf{r}(t)=\left\langle t^{2},-t^{2},-3 t\right\rangle$ with $0 \leq t \leq \frac{1}{2}$. Note: I purposely made it impossible to compute the line integral in the usual way. You must use the Fundamental Theorem of Line Integrals to solve this problem properly.

Step 1 ：Given the scalar function \(f(x, y, z) = e^{xyz} + \arccos(x+y) - z^2\) and the vector function \(\mathbf{r}(t) = \langle t^2, -t^2, -3t \rangle\) with \(0 \leq t \leq \frac{1}{2}\).

Step 2 ：The Fundamental Theorem of Line Integrals states that if a vector field F is the gradient of a scalar function f, then the line integral of F along a curve C from point A to point B is equal to f(B) - f(A).

Step 3 ：In this case, the vector field F is the gradient of the scalar function f. The curve C is parameterized by the vector function r(t).

Step 4 ：The points A and B are the initial and final points of the curve C, which are r(0) and r(1/2), respectively. We can substitute the coordinates of points A and B into the scalar function f to compute f(B) - f(A).

Step 5 ：Substituting the coordinates of points A and B into the scalar function f, we get \(f(A) = f(0, 0, 0)\) and \(f(B) = f(0.25, -0.25, -1.5)\).

Step 6 ：Computing the difference \(f(B) - f(A)\), we get the value of the line integral as -2.1517148596921745.

Step 7 ：Final Answer: The value of the line integral is \(\boxed{-2.1517148596921745}\).