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

irrational, then, just means all the numbers that aren't rational. every integer is a rational number, since each integer n can be written in the form n/1. however, numbers like 1/2, 45454737/2424242, and -3/7 are also rational, since they are fractions whose numerator and denominator are integers.

step-by-step explanation:

5

step-by-step explanation: