2008级985高等数学（下）及其参考答案 成的区域的整个边界曲面.
解：S?S1?S2.其中 S1:z???z???z???1???dxdy??????x???y?22x?y,
22 ，
yx?y22dS??1????xx?y2?22?????????122??dxdy???22dxdy.
???xS12?ydS?2?2??x?ydxdy?Dxy?22?2?20d??r.rdr?022?;
S2:z?1, dS???z???z?1???????y??dxdy?dxd.y， ?x????2
???xS12?y?dS?2???xDxy22?y?dxdy?2?2?0d??r.rdr?01212?;
所以
???xS2?ydS?????xS12?y2?dS????xS12?ydS?2?12?1?2?.
??5．求幂级数?n?1?1n?n?1??xn在x???1,1?内的和函数.
?解：?n?11?n?1x?????x?n?n?1?n?1?n?1?nn1?nxn?11n?1?n?1x
nn?1? 设s1?x????n?1x??xn?? 则s1??x?????n??nn?1??n???n?1xn?1?11?x.
所以， s1?x??s1?0????x11?x0dx??ln1?x.
?即
??nxn?11n??ln1?x 设s2?x??x1?x??n?1xn?1n?1
??xn?1???x????? 则s2?n?1??n?1?????n?1nxn?. 则
s2?x??s2?0???xx1?x0dx??x?ln1?x.
故
?n?1xn?1?1?1?xxn?1n?1n?1?1xs2?x???1?ln1?xx.
所以
?n?n?1?n?11xn=?ln1?x?1?ln1?xx.
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2008级985高等数学（下）及其参考答案 6．计算曲面积分??x2ydzdx?y2zdxdy，?是柱体
?????x,y,z?|x????2?y2?a,0?z?h2?的外侧.
解：原式?2??x2y?yz????dxdydz
?z???y?????????x?2?y2?dxdydz??2?0d??rdr0a?h0rdz?h?22?0d??rdr?0a3?2ha.
47．（本题有两小题，周六课时的同学都做，周五课时的同学任选一题） （1）计算三重积分?????x2?y2?z2?22,zdxdydz,其中区域?是由?2所确定. 222?x?y??z?2??22（2）设函数f?u?具有连续导数且f?0??0,求 lim1t?0?t42???2f2?2x?y?z222?dv.
x?y?z?t解：（1）解法一：柱面坐标法：
?x2?y2?z2?22,联立?2消z,得?在xoy坐标面上的投影区域为 222??x?y?z?2?2,?圆域D:x2?y2?3.
2?34?r????22zdxdydz
2 ??0d??30rdr?2?4?rzdz?2?2?30?z3r.??3?3|4?r2??dr 24?r??2 ?2?32?32?34?3??0?r???4?r2???2?334?r2????dr（令r?2sint）
??30 ?2sint8cost??2?2cost?.2costdt
3?? ??30322cost?1?3cost?3costsintcostdt
?32???.32?costsintdt?3042?3?.32?3costsintdt
0?320?03tsintdt ?2?.32?costsintdt?2?.32?3cos ??
4?3.32cost55?|30?2?3.32cost22?|30
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2008级985高等数学（下）及其参考答案 ?2?.324?cost33?|30?2?.32cost44?|30
?? ?2??31???3??7???15?.32??.32????.32????.32????? 15332?32??8??4??16?59?59?.32?48015
其实，此题最宜采用球面坐标计算：这时首先要把积分区域?分成两个子区域： ???1??2.
?0???2?,?0???2?,其中 ???:??2??1:?0???, ?2?33???,
?0???2,?2?0???4cos?,则
???z2dxdydz=???z2dxdydz????z2dxdydz
??1?22????2?2222cos?0d??30sin?d??0?cos?.?d???2?0d???sin?d??4?2cos20?.?2d?
3?2?????3sin?.cos2?d????2????4d????2????224cos?4???sin?.cos?d???d???0???0???0? 3???2??????1cos3?3???152??1?3|0????|0???2???5?232sin?.cos7?d????5?32.? ?3??2?.7164?24.5.32?5.32??1?8??2???8cos?|?? 3??7?60.32?64??7??59?5.32?1.1??2568???60.32?160.32?15. （2）
f?x2?y2?z2?dv?2???22?t2?2??0d??sin?d??tf????200d?
x?y?z ?4??tf????20d?.，
所以，4?lim1t?0?t42???f?x2?y2?z2?dv?lim?t0f????2d?222t?0x?y?z?t?t4（洛必达）
2 ?l4?f?t?.tf?t??limf?t??f?0?t?im04?t3 ?limt?0tt?0t?0?f??0?.
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