Question

Evaluate the line integral ∫C F · dr, where F = ⟨2sin(x_1) - 4cos(y), xz⟩, and C is the path given by r(t) = ⟨t⟩.

Answer 1

The student is asked to evaluate a line integral where the path appears to be incorrectly or incompletely given. A proper path parametrization is necessary to carry out the integration.

Step-by-step explanation:

The student has asked to evaluate the line integral ∫C F · dr, where F = ⟨2sin(x_1) - 4cos(y), xz⟩, and C is the path given by r(t) = ⟨t⟩. To evaluate a line integral, one typically parametrizes the path C and then computes the dot product of the vector field F and the differential dr along the path. If r(t) only contains one component, it implies a motion along one axis. However, the vector field F, in this case, has three components (in terms of x, y, and z), so the parametrization of r(t) as given by the student appears to be incomplete.

There might be some mistake or misinterpretation in formulating the path r(t). In a proper setup, after parametrizing the path, one would substitute the parametric equations into F and then integrate along the path using the given parameter, usually t.