二重积分的计算方法_图文.ppt

1:二重积分累次积分(即两次定积分).2,( , )d ( , )d dD Df x y f x y x y???? ??d d dx y??故二重积分可写为xyoDD则面积元素为一、利用直角坐标系计算二重积分3(2)如果积分区域为:其中函数、在区间上连续.)(1x?)(2x?],[ba[X-型]b)(2xy??)(1xy??aDxOyxOy)(1xy??)(2xy??Dba,bxa??).()(21xyx????4:平行截面面积为已知的立体的体积xoxdxx?ab)(xA表示过点x且垂直于x轴的截面面积,)(xA为x的已知连续函数d ( )dV A x x?( )dbaV A x x??立体体积)(xA此方法关键是求5)0),((d),(???yxfyxfD??计算截面面积),(yxfz?( 红色部分即A(x0) )*以D为底,“平行截面面积为已知的立体求体积”:是区间)](),([0201xx??为曲边的曲边梯形.),(0yxfz?为底,曲线xyzO),(yxfz?D)(2xy??)(0xAab0x)(1xy??6),(yxfz?D)(2xy??)(0xAab0x)(1xy??yzO)(01x?)(02x?),(0yxfz?A(x0))(01x?)(02x?yyxfxAd),()(00??yyxfxAxxd),()()()(21???????DyxfV?d),(??baxxAd)(xbad??)d),(()()(21?xxyyxf?????baxxyyxfx)()(21d),(d??先对y后对x的二次积分(累次积分)7(2)积分区域为:,dyc??)()(21yxy???????Dyxf?d),(先对x后对y的二次积分也即??dcyyxyxfy)()(21d),(d?????Dyxf?d),(其中函数、)(1y?)(2y?],[)(2yx??cd)(1yx??xOyxOyD)(2yx??cd)(1yx???dcyd)d),((?xyxf)(1y?)(2y?[Y-型]8-型区域也不是Y-型区域时,将D分成几部分,使每部分是X-型区域或是Y--型区域也是Y-型区域时,可以用两个公式进行计算.ⅠⅡⅢyx0yx0cdabD9?????Dbadcyyxfxyxfd),(dd),(?)()(),(21yfxfyxf??若???yxyfxfDdd)()(:则则a≤x≤b,c≤y≤d??baxxfd)(1yyfdcd)(2??yyfxfdcd)()(21??xd)?ba(???dcbaxyxfyd),(d10,:>,,下限应为常数(后积先定限)..