Chapter 17.2, Problem 81E

Looking ahead: Area from line integrals The area of a region R in the plane, whose boundary is the closed curve C, may be computed using line integrals with the formula

     $areaofR={\displaystyle \underset{C}{\int }xdy}=-{\displaystyle \underset{C}{\int }ydx.}$

These ideas reappear later in the chapter.

67.    Let R be the rectangle with vertices (0, 0), (a, 0), (0, b), and (a, b), and let C be the boundary of R oriented counterclockwise. Use the formula A = ò_C x dy to verify that the area of the rectangle is ab.