平面图形面积计算公式的注记

A Remark on Calculation Formulas of Plane Area

华东师范大学数学系编《数学分析》上册第245页在处理极坐标方程表示的曲线所围区域的面积时,采用对角度分割区域、用小扇形近似和的极限得到了面积的计算公式,然而在下册第224页平面区域面积的定义时,采用平行于坐标轴的直线网分割区域、用小矩形近似和的极限来定义面积.这两种定义的一致性是不显然的,本文将正面回答这一问题,进而对该教材中的几个面积公式给出详细说明.

A formula of the area surrounded by the curve of polar coordinate equation was obtained in pages 245 of“mathematical analysis”written by Department of mathematics,East China Normal University,where the region was divided by the argument,and approximated by the sum of the small sectors.However,the area of planar region was defined in pages 224,where the region was divided by the straight lines paralleling to the axis,and approximated by the sum of some small rectangles.It is nature to ask if both of two methods are consistent.In this paper,we will give the positive answer.