第三章一元函数积分学及其应用第一节定积分概念与性质P178:A:2,4,6,8(双号),10,12,14;B:4,5,6.axyo)(xfy1ixixi))((1iiixxf1x2x2nx1nx1.1定积分问题举例bxaxyxfy,,0),(b例1.1曲边梯形的面积问题分011iinaxxxxxbiiixxx1ni,,2,1匀1iiiixxni,,2,1iiixfS)(合niiinxfSS1)(精niiixnxxfSSii10max0max)(limlim设函数)(xfy在区间[a,b]上有定义,经(1)分011iinaxxxxxbiiixxx1ni,,2,1(2)匀i1iiixxiiixfS)(1.2定积分的定义(3)合niiinxfS1)((4)精niiixnxxfSii10max0max)(limlim如果该极限存在,则称此极限值为函数)(xf在区间[a,b]上的定积分,记作badxxf)(,即badxxf)(niiixxfi10max)(lim1.3定积分存在的条件badxxf)(niiixxfi10max)(lim()iiimfM111()nnniiiiiiiiimxfxMx10,0,||,|()|iniiiixMmx当充分必要条件oxyab1,iixx()yfx充分条件定理如果被积函数)(xf在积分闭区间],[ba上连续,或者仅有有限个第一类间断点,则定积分badxxf)(必然存在。1.3定积分存在的条件必要条件有界可积可积连续10,0,||,|()|iniiiixMmx当oxy12xyn1n2ni1ninn1用定义计算定积分102dxx例1.2解连续必可积211()nkknn31(1)(21)6nnnn12013xdx1.4定积分的性质abba)1(0aababaxxfkxxkfd)(d)()3(bccaba)2(bababaxxgxxfxxgxfd)(d)(d)()()4(niiiixxgfi10max)()(limniiixniiixxgxfii10max10max)(lim)(lim(5))()(,xgxfba0)()(lim10maxniiiixxfgibabaxxfxxgd)(d)(Mxfmba)(,)6(baxxfbad)()7(babaxxgxxfd)(d)(xxfxgbad))()((baabMxxfabm)(d)()(xxfbad)(bababaxMxxfxmdd)(diniiiniixfxf11)()(niiixf1)((8)中值定理],[],[)(bacbaCxfMabxxfmbad)()(d)(],[cfabxxfbacba)()()(abcfdxxfba的介值定理由连续函数证],[)(baCxfMxfm)(xMxxfxmbababadd)(doxyabc)(cfy(9)广义中值定理],[),0(0)(],,[)(),(bacxgbaCxgxfxxgcfxxgxfbabad)()(d)()(证],[)(baCxf)()()()(xMgxgxfmxgxxgMxxgxfxxgmbababad)(d)()(d)(Mxxgxxgxfmbabad)(d)()(xxgxxgxfcfbacbabad)(d)()()(],[Mxfm)(0)(xg例1.3用定积分定义计算xexd10解]1,0[Ceyxxexd10oxy11.ixnnnninni1,,1,,10)2(2111(),iniiiSfxen1113210)1(nnnininnnneneiniinninii11)3(11111nueennn1111lim)4(1neen1111ueeuu11lim01eix1ix例1.4用几何意义证明定积分)(2d)1(22abkxkxba2d)2(222axxaaa解xkxbad)1(oxyab21)()(abbakxxaaad)2(22oxya212a22xay例1.5证明不等式21d152)1(212xxxexexd1)2(102解2121152)1(2xxx21)(xxxf0)1(21)(2222xxxxf2121221d21d1d52xxxxx21d152212xxx101)2(2xeexexexd1102例1.6判别积分的大小11200(1)xdxxdx与2200(2)sinxdxxdx与解2,10)1(xxxdxxdxx10210xxxsin,20)2(dxxdxx2200sinoxy2xyxysin