Chapter 14.2, Problem 68E

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.

68.    Let R = {(r, θ): 0 ≤ r ≤ a, 0 ≤ θ ≤ 2π} be the disk of radius a centered at the origin and let C be the boundary of R oriented counterclockwise. Use the formula A = –ò_Cy dx to verify that the area of the disk is πr².