1. Evaluate the following line integral, where $C$ is the line segment from $(1,4)$ to $(0,2)$. \[ \int_{C} \sin (\pi y) d y+y x^{2} d x \]

Step 1 ：Define the vector field F(x, y) = [y*x^2, sin(pi*y)].

Step 2 ：Define the curve C as the line segment from (1,4) to (0,2).

Step 3 ：Parameterize this line segment by a function r(t) = [1-t, 4-2t] for t in [0, 1].

Step 4 ：Calculate the differential of the position vector dr = [-dt, -2dt].

Step 5 ：Substitute these into the integral and evaluate.

Step 6 ：\( t = t \)

Step 7 ：\( x = 1 - t \)

Step 8 ：\( y = 4 - 2*t \)

Step 9 ：\( dx = -1 \)

Step 10 ：\( dy = -2 \)

Step 11 ：\( F = \begin{bmatrix} (1 - t)^2*(4 - 2*t) \\ sin(\pi*(4 - 2*t)) \end{bmatrix} \)

Step 12 ：\( dr = \begin{bmatrix} -1 \\ -2 \end{bmatrix} \)

Step 13 ：Calculate the integral to get the final answer.

Step 14 ：Final Answer: \(\boxed{-\frac{7}{6}}\)