?x?f??x?dx??xd?f?x???x?f?x???f?x?dx?十三、二重积分
x1?xD2?lnx?1?x2?Cx?y??yx34、求0?dx?1xsinydy y0x?y21?x?y?x解：由题意知，积分区域D为?
0?x?1?x?dx?01xx1ysiny1sinysinydy??dy?2dx??y?y2dy0y0yyy
???????1?y???siny??dy???y?1?d?cosy???y?1?cosy0??cosydy
1000111?1?siny0?1?sin1
35、计算二重积分I1??dx?sinydy
0x112yDx?y1?x?y?1解：积分区域D为?
0?x?1?1121y201x11I??dx?sinydy??dy?sinydx??2ysiny2dy
0x002011112221??sinydy?cosy??1?cos1?
02022十四、无穷级数
1131?1??1?3??， ?????
36、???111?321?32n?0?3?n?1?3???nn117?11??1? ?n?n???12?1?31?22n?0?3?11
1??1???的敛散性 37、判断??1???nn?1?n?1n?1111u?????un?1即?un?? 解：nnn?1n?n?1??n?1??n?2??limun?0
n??1??1????1??n?n?1?收敛。
??n?1n?1??1??1????的敛散性，如果收敛是绝对还是条件。 38、判断??nn?1?n?1n??1??11?1n??1??????????1?解：?
????nn?1nn?1nn?1??n?1n?1n?1n?11?n?n?1??n2?1，又?2n?1n?收敛
1??n?n?1?也收敛
n?11??1????1??n?n?1?绝对收敛。
??n?1nnax39、求?nn?0??的收敛域及收敛半径R
1n??○
liman?1xn?1anxn记an?1?lim?x??x n??an12
2当?x?1即○
x?1?1nax时，?nn?0??收敛；
当
?x?1即
1x??nax时，?nn?0发散；
?11??11??11??11???,?、??,? ?,??,?x??时，收敛域可能是?当、?、?????????????????????收敛半径
?R?1?
?1112n?1x?n40、求?的收敛半径及收敛域、和函数，并求?n?02n?12n?02n?1的和。
解：
?n??limu2n?1u2n1x2n?32n?122n?3?lim?limx?x2n??n??2n?3 12n?1x2n?1?当
x?1即
212n?1xx?1时，?收敛；
2n?1n?012n?1xx?1时，?发散；
2n?1n?0??当
x?1即
212n?1x当x?1或x??1时，?发散；
2n?1n?0?收敛域为??1,1?，收敛半径R?1。
?11?12n?1?2n2n?1S??x????xS?x???x????x??2记，则
2n?11?x2n?1?n?0n?0?n?0???
13
??0x11a?11dx?ln?C S??x?dx??dx，?2201?x22aa?xa?xx即
S?x?0x11?x?ln21?xx，又S0?0??0
?11?x111?x2n?1?ln,x???1,1? ?S?x??2ln1?x,x???1,1?即?2n?1x21?xn?0111?1??n?????n?02n?12n?02n?1?2?1?1??2?ln?2??1?1212??2n?11?1??2?2???n?02n?1?2??2n?1
???2ln?2???2?12?1???22?1??2ln2?1
??141、x?1是arctanx?1的 第一类 间断点。
1?1?arctan??，limarctan?解：lim??x?1x?12x?1x?1222??D?x,yx?y?2x42、设
y????1dxdy??，则D
0?n2xn?1??1?n的收敛性
43、判断?3n?2n?1?n?1limun?1????1?3n??3n，n??un解：
n?2n?2?nn?2n?11n?213?lim?lim???1n??n?1n??3n?13
3nn?1??1?n收敛。
??3n?2n14
??x?acostd2y44、设?求2y?bsintdx?dy?解：dx
yt?bcosta???cott xt??asintbd2yd?dy?????? 2dxdx?dx?45、设解：对
2f?x?可导且满足?0x?f?t?dt?f?x??x求f?x?
xx求导，x?f?x??f??x??2x即f??x??x?f?x??2x
?P?x?dx???fx?e?x22?Q?x?e?P?x?dxdx?C??e????x?dx?2xe???x?dxdx?C? ??????????22xx2?????x2??e??2??x?e2d???C?e??2???????xx?????2e2?C???2?Ce2????22
又
f?0??0，得C?2
x22?f?x???2?2e46、若
1?23?10f?x?dx?2，则?x0f?x?dx?63
47、已知
?f?t?dt?x03x32?1则
2f?1??
3解：对
x两边求导，
48、制作一容积为
??V?m?2fx3?3x2?2x
的无盖圆柱形桶，底用铝板制，侧壁用木板制。已知每平方米铝板价是木板价
的5倍，问桶底圆的半径R和桶高h为多少时，总费用最少？ 解：设木板价为P元/m，则总费用F?5P?R2?P?2?R?h
2V又V??Rh，得h??R222PV，则F?5P?R?R15
hV
R??10P?R??FR2PVR2V??0得R?，令FR5?3
又F的最小值一定存在。
V?当R?5?3，h?5R?325V?时，F最小。
49、y?1?1?x2x的定义域为
??1,0???0,1?
16