The Integral Theorems

1. Let C be the curve in R2 consisting of line segments from (4, 1) to (4, 3) to (1, 3) to (1, 1). Let

~F (x, y) =⟨x+ y, (y − 1)3esin y

⟩. Evaluate the line integral

∫C

~F · d~r.

1 2 3 4x

1

2

3

4y

2. Let C be the (oriented) curve parameterized by ~r(t) = 〈cos t, sin t, t〉, 0 ≤ t ≤ 2π. Let ~F (x, y, z) =

〈ex2

, (sin y + 3)y, z2〉. Evaluate∫C

~F · d~r.

1