Question

Find the line integrals of F equals 4yi plus 3xj plus 4zk from (0, 0, 0) to (1, 1, 1) over each of the following paths.

A) Straight line path
B) Circular path
C) Parametric path
D) Polygonal path

Answer 1

To find the line integral of a vector field over different paths, we can use parameterizations of each path. For the straight line path from (0, 0, 0) to (1, 1, 1), the line integral is 11/2.

Step-by-step explanation:

To find the line integral of the vector field F = 4yi + 3xj + 4zk over different paths, we can use different parameterizations of each path. Let's go through each path:

A) Straight line path:

The straight line path from (0, 0, 0) to (1, 1, 1) can be parameterized as r(t) = ti + tj + tk where t ranges from 0 to 1.

The line integral is given by the formula ∫01 (F⋅dr) = ∫01 (4t + 3t + 4t)dt = ∫01 (11t)dt = [11t2/2] from 0 to 1 = 11/2.