Consider the functions below. f(x, y, z) = x i − z j + y k r(t) = 4t i + 6t j − t2 k (a) evaluate the line integral c f · dr, where c is given by r(t), −1 ≤ t ≤ 1.

Applying the steps, the result of the line integral is -4.

The curve is:

The vector field is:

Applying the vector field at the curve, we have that:

The derivative of the vector field is:

The dot product of the vector field along the curve with the derivative is:

Hence, the line integral is:

A similar problem is given at link

With

we have

The vector field evaluated over this parameterization is

so the line integral is

i would say the third one.

step-by-step explanation: liner would mean in a line and the only one that doesn't look like it's even thinking about forming a line would be the third one