高等数学(北大版)答案习题3.5

1习题3.213212002.,1.43(2).22xyxyyySyydyy与解222232132313.211.21(1)21,140,0,1;4,3.11(1)23116.2263yxxyyxxxxyxxxyxySyydxyyy与解2222225.42.442,2yxyxxyxxxxyxx与解2224221123/2200:1..,,(1)(1)0,0,1.211().333yxxyyxxxxyxxxxxxSxxdxxx求下列曲线所围成的的图形的面积与求交点解:2022202242400/224202(sin)4.002(a0)(1cos)(1cos)(sin)(1cos)4sin8sin2316sin164223.xattytyatSatdattatdttadtauduauduaa与2212122212213222240,(22)(2)0,2,1.(24)(224)249.3xxxxxxSxxxdxxxdxxxx2222224242222221222212222022016.8().28181424320,4320,(8)(4)08()4,4,2,218212824284arcsin22322xyyxxyxxyxxxxuuuuuuuxxxSxxdxxxdxxxxS与分上下两部分舍解244826.3322221211222213227.42.442220,(2)(1)0,2,1.(42)6()96.322yxyxyxxxyxxxxxxxSxxdxxxdxxx与解322/422/42008.cos2(0).1cos22sin2|.2raaadaa其求双纽线所围图形的面积1解S=422/32/32/333/2226200/23720/237203:9.(0).cos,02.sin22sin3cossin6sincos6sin(1sin)6428326175391axyaaxattyatVydxatattdtattdtattdta求下列曲线围成的平面图形绕轴旋转所成旋转体的体积解3.05aln3ln32200ln32010.1,ln3,.(1)(21)12ln3.2xxxxxxyexyeVedxeedxeex231/32/31/32/3202/37/32/37/30:11.,0(0,0).,()33.77bbayxxybabxayVaydyayab求下列平面曲线围成的平面图形绕轴旋转所成旋转体的体积及解413.()[,](0)()2().2(),2().babayfxabayfxxaxbyVxfxdxdxdVxfxdxVxfxdx设在区间上连续且不取负值，试用微元法推导：由曲线，直线，及轴围成的平面图形绕轴旋转所成立体的体积为厚度的圆筒的体积解2111111121118ln12.,0,.8ln8ln8ln|ln1812881.eeeeeeyxxyeyyVdyyydyyyydyydyeyee解2211222211122214.,1,22222222()2.xxxxxxyexxxyVxedxxdexeedxeeeeeeee求曲线及轴所围成的平面图形绕轴旋转所成的立体的体积.解52222223233215.:.3(),.1()|31(())33aaahahahhVhayfxaxahxaVaxdxahxhahaahha证明半径为高为的球缺的体积为证3236420320/22017.sin.3sincos,33sinsincos333sin3136sin6.222rarasadadadaa求曲线的全长解3322/22/20018.cos,sin.3cos(sin),3sincos143sincos12sin|6.2xatyatxattyattsattdxata求向星形线的弧长解322223213433211116.13.621.2211221114.2623xyxxxxyxxsdxxxxdxxx求曲线在到之间的弧长解61224020.2cos2(0)42.1dxraaLLax试证双纽线的全长可表为202222032/4250022222221.1(02)4.2211421444cossecsectansectan(2)sectansectan(2)(2),1nnnnnnnnnxyxxxyxxSdxdxxdxxIxdxxdxxxnxxdxxxnInIIn求抛物线绕旋转所得的旋转体的侧面积.解222sectan.11nnnxxIn2200019.(1cos).(sin)2(1cos)sin4cos8sin8.22rarasadadxaa求心脏线的全长解2242/4220/40/4220/422220/4440/44024sin2,2sin2/,4sin242cos22cos2142cos242cossin42(cossin)(cossin)42cossintan42(tan)1tan42rrararasadaaddadadadaxdxa证140.1x7335313131311sectansectansectan444422133sectansectanln(tansec).488IxxIxxxxIxxxxxxC3/401334(sectansectanln(tansec))|488[723ln(12)].2Sxxxxxx2222/222220/222220122220122220222.(0),.cos,02,sin,cos.sin22sincossin4()coscos4()44axybaabxattxatybtybtSbatbttdtbaabtdtbaabudubabuduaab求1分别绕长短轴旋转而成的椭球面的面积解1222202/222220/22222012222021222220222222222arcsin22arcsin21.22sincoscos4()sinsin4()44ln22()bbuuuaabSaatbttdtababtdtababudubaabuduabubbaabuabab12222022()22ln(1).buuabbaab22223.(,0)xyaahyahay计算圆弧绕轴旋转所得球冠的面积.8101020000025.10m()(70.2)(70.2)70.180(kg).26..cos0.,0sinsin2[cos]|.2(0,).27.,xxmxdxxxaxattyatatadtaaytaa0有一细棒长已知距左端点x处的线密度是kg/m求这细棒的质量.求半径为的均匀半圆周的重心坐标由对称性,x重心坐标有一均匀细杆解解/54/52200../5./.lllMlMMMlJxdxxdxll长为质量为计算细杆绕距离一端处的转动惯量解/54/53320013.3375llMxMxMlll222arcsin22arcsin22arcsin2cosarcsin.sin222cossin212.ahaahaahaxatahtyataSxxydtatdtatahaaha解aha222223/23/22123/21125/21224.(1cos).2(1cos)sin(1cos)sin22(1cos)sin221coscos22(1)222(1)532.5raSaaadadadaxdxaxa求心脏线绕极轴旋转所成的旋转体的侧面积000解r=-asin.922242222002223228.,,,.2.2.221.4229.,,,,33,,13aaaMMMMxdxdmxdxaaaMxdxMxJxMaaaMahaMMaMyxdmxdxxdxhahhhahd设有一均匀圆盘半径为质量为求它对于通过其圆心且与盘垂直的轴之转动惯量有一均匀的圆锥形陀螺质量为底半径为高为试求此陀螺关于其对称轴的转动惯量.解=解2245225425500552001132213133.2251030.,2kg/m.29.8.29.89.8259.8().hhaaMJdmxxdxhhaMaMxJxdxMahhdWxdxWxdxxJ楼顶上有一绳索沿墙壁下垂该绳索的密度为若绳索下垂部分长为5m,求将下垂部分全部拉到楼顶所需做的功.解2231.()[,],,(),,,(),(),().32.48m,64m,164,06424,,9bayfxabyfxxaxbxydSfxdxdFpdSgxfxdxFgxfxdxyaxaax设在上连续非负将由及轴围成的曲边梯形垂直放置于水中使轴与水平面相齐求水对此曲边梯形的压力.一水闸门的边界线为一抛物线,沿水平面的宽度为最低处在水面下求水对闸门的的压力.解解642082083503(64).664.64,64,08,64,0.6(64)(2)126452428.8.35yFgyydyyuyuyuyuFguuudyuugg时时6424