Question

Let C be the line segment from (1,1) to (4, -3) and F=x+y> evaluate the integral over C of (F•T)

Answer 1

To evaluate the given line integral over the line segment from (1,1) to (4, -3), we can parametrize the line and substitute it into the vector field. Integrating the resulting expression will give us the value of the line integral.

Step-by-step explanation:

To evaluate the given line integral, we first need to parametrize the line segment. Let's parametrize the line segment C as r(t) = (1 + 3t, 1 - 4t), where t ranges from 0 to 1. Now, we can evaluate the integral by substituting this parameterization into the vector field F = x + y.

So, F · dr(t) = (1 + 3t) + (1 - 4t) = 2 - t. Now, we can set up the integral:

1. ∫01 (2 - t) dt.
2. Integrate with respect to t: [(2t - t²/2)]01 = (2 - (1/2)) - (0 - 0) = 3/2.

Therefore, the value of the line integral over C of (F · T) is 3/2.