Advanced Calculus fi..

Also the values y:, , y:2, . , . satisfyChapter 4 Integral Calculus of Functions of Several Variables 263If we combine (4.99) and (4. loo), we obtainOn the right-hand side we have a sum like that for the double integral; on theleft-hand side we have a constant c. It follows that as the mesh h + 0, the sums onthe right-hand side have limit c.However, the sum in (4.101) includes terms for rectangles not wholly containedin G. This is clear from Fig. 4.16, where the top and bottom rectangles extend beyondG. We show below that for given r, a 6 > 0 can be found so that for 0 < h < 6 theabsolute value of the sum of such terms is less than r/2. Therefore we can write, forO