Question

Evaluate the line integral ∫ c f ⋅ d r where f = Ÿ¨ - 4 sin ¡ x , - 2 cos ¡ y , x z Ÿ© and c is the path given by r ( t ) = ( - 3 t 3 , 3 t 2 , 3 t ) for 0 ≤ t ≤ 1. What is the value of ∫ c f ⋅ d r?

Answer 1

To evaluate the line integral, compute the derivative of the path, substitute into the vector field, find the dot product, and integrate with respect to the parameter from 0 to 1.

Step-by-step explanation:

To evaluate the line integral ∫ c f · dr where f = ⟨-4 sin x, -2 cos y, xz⟩ and c is the path given by r(t) = (-3t³, 3t², 3t) for 0 ≤ t ≤ 1, we need to follow a few steps.

Find the derivative dr/dt of the path r(t).

Substitute the parameterized path r(t) and dr/dt into the vector field f.

Compute the dot product f · dr/dt.

Integrate the result with respect to t from 0 to 1 to find the value of the line integral.