Chapter 14.7, Problem 40E

Area of a region in a plane Let R be a region in a plane that has a unit normal vector n = 〈a, b, c〉 and boundary C. Let F = 〈bz, cx, ay〉.

a.    Show that ▿ × F = n

b.    Use Stokes’ Theorem to show that

$\text{area}\text{of}R={\displaystyle \underset{C}{\oint }F•dr.}$

c.    Consider the curve C given by r = 〈5 sin t, 13 cos t, 12 sin t〉, for 0 ≤ t ≤ 2p. Prove that C lies in a plane by showing that $\text{r}\times {\text{r}}^{\prime }$ is constant for all t.

d.    Use part (b) to find the area of the region enclosed by C in part (c). (Hint: Find the unit normal vector that is consistent with the orientation of C.)