Question

Evaluate the line integral of f(x, y, z)=x-3y²+z over the curve formed by the union

Answer 1

To evaluate the given line integral, you need to parametrize the curve and then integrate the resulting expression over the parameter

Step-by-step explanation:

The line integral of f(x, y, z) = x - 3y² + z over the curve is given by the formula:

∫f(x, y, z) ⋅ ds

To evaluate this line integral, we need to parametrize the curve.

1. 1. Let's assume that the parameterization of the curve is given by r(t) = (x(t), y(t), z(t)), where a ≤ t ≤ b.
   2. Then, the line integral becomes:

∫(x(t) - 3y(t)² + z(t)) ⋅ ||r'(t)|| dt

To compute ∫r'(t) ⋅ ||r'(t)|| dt, we can use the chain rule and compute ||r'(t)|| dt in terms of t. We then integrate the resulting expression for t from a to b.