1?????????a?2?z?a???dxdydz?a?????adxdy?， D?2其中?为?+S?围成的空间区域，D为z?0上的平面区域x2?y2?a2，
?1?2322于是I???3a??a?2???zdxdydz?a?a?
a?3?? ?1??a?a4?2?2?0d??a0dr?0?a?r22zdz?
???2a3.
解法二. 直接分块积分
I1?1a??axdydz? ??2??a2??x2?y2?dydz，
Dyz其中Dyz为yOz平面上的半圆y2?z2?a2，z?0. 利用极坐标，得
I1??2?2??d??a0a?rrdr??22223?a，
3I2?1a???z?a?dxdy?
22 ?1?a?????aDxya??x?y2????2dxdy，
其中Dxy为xOy平面上的圆域，x2?y2?a2， 用极坐标，得
I2?1?a?62?0d??a0?2a?2aa?r?r2222?rdr
?a3，
?2a3因此I?I1?I2??.
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（3）现要设计一个容积为V的圆柱体的容积，已知上下两低的材料费为单位面积a元，而侧面的材料费为单位面积b元.试给出最节省的设计方案：即高与上下底面的直径之比为何值时，所需费用最少？ 解：设圆柱体的高为h，底面直径为d，费用为f，
2根据题意，可知??d???h?V2?2?，dh?4V?
?d2f?a?2??????b??dh?2?
????1ad2?bdh??2??
???1?ad2?11??22bdh?2bdh??
??13223ad?bdh?bdh
?3?3ab2?3?d2h?22 2?3?32ab2?3?4V??，
????当且仅当ad2?bdh时，等号成立，
hd?ab，
故当hd?ab时，所需要的费用最少.
（4）已知f?x?在?11?,?1?42??内满足f??x??sin3x?cos3x求f?x?.
解：f??x???1sin3x?cos3xdx
?2?11?3?x?sinx?cosx??sinx?cos2sin2x?sinxcosx?cos2x?dx?
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，?sinx?cosx1dx?1?21???sin?x??4??12dx
x?lntan2?4?C，
1 ??sin?2?sinx?cosx2x?sinxcosx?cosx2dx??12sinx?cosx?sinx?cosx?2?12dx
sinx?cosx?sinx?cosx?2dx?1
?2?d?sinx?cosx??sinx?cosx?2?1
?2arctan?sinx?cosx??C2
2132x?lntan?所以，f?x??4?2arctansinx?cosx?C??23.
求下列极限.
n???1? （1）limn??1???e?；
???n??n???
11?1?nnna?b?c? （2）lim??n???3????n，其中a?0，b?0，c?0.
nx??????1?1?解：（1）limn??1???e??limx??1???e?
???x??????n??n?x????? ?lime?1?xln?1??x???ex???1x
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1???1?1????11?ln1??x????????x??x1?xx????????limx???1?2xx
1?1?ln?1???x?1?x??elimx???1?2x11?x?1x2?1
?1x31 ?elim?1?x?2x???
?e2x???lim?1?x?1x22
e2 ??e2x???lim11??1???x??1x2??.
?n??x?nnxxa?b?ca?b?c??lim??lim???n???x????33????????111n11
1x1x11x1x1lnlimx???a?b?cx31x ?limex???xlna?b?cx3 ?e，
111lnx???ax?bx?cx31xlim
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11?1??1?xxxalna?blnb?clnc???2?111???x?xxx?a?b?c?lim
x???1?2x111?1?xxx?lim1alna?blnb?clnc? 11?x????ax?bx?cx?1?13?lna?lnb?lnc??ln3nabc，
11?1?nnna?b?c?故lim???n???3????3abc. ??一般地，有lim?n???????a?k?1??m???mnk11nma1a2?am，其中ak?0，k?1,2,?,m，
?ex?e2x???enx?xelim???limx?0x?0n??lne?elne?ex2x???enxnx
?2e2x?x2x???exnx??lnn1?limex?0?ex?0n?12lime?ex2x???enx?ex???nenx?1
1?en?1?2???n??e.
设f?x?在x?1点附近有定义，且在x?1点可导，f?1??0，f??1??2， 求limf?sinx?cosx?22x?0x?xtanx2.
解：limf?sinx?cosx?x?xtanx2x?0?sin2x?cosx?1f?sin2x?cosx??f?1??? ?lim??22x?0??x?xtanxsinx?cosx?1?? 15