Question

Evaluate the line integral using the fundamental theorem of line integrals.

∮C (4y dx + 4x dy), where C is the line segment from (0, 0) to (2, 2).

a) 16
b) 8
c) 4
d) 2

Answer 1

To evaluate the line integral, we need to parameterize the line segment and substitute the values into the line integral formula. The value of the line integral is 8.

Step-by-step explanation:

To evaluate the line integral using the fundamental theorem of line integrals, we first need to parameterize the line segment from (0, 0) to (2, 2). Let's choose the parameterization r(t) = (2t, 2t), where t ranges from 0 to 1. We can calculate dx and dy in terms of dt as follows:

dx = 2dt

dy = 2dt

Now, substitute these values into the line integral formula: ∮C (4y dx + 4x dy)

= ∫01 (4(2t))(2dt) + 4(2t)(2dt)

= ∫01 16t dt

= 8[t2]

= 8(12 - 02)

= 8

Therefore, the value of the line integral is 8. Therefore, the correct answer is (b) 8.