（完整word版）微积分（数学分析）练习题及答案doc

?? 2? 0d?? 3 0?1?r?z2?2?2? r3 4?r2 3?1 2?r5?3dr??d???4r?r??dr.

02 09???214r6?113??2??2r?r????. 2454? 04?46.解：因为z?a2?x2?y2,故z?x??2x2y 3xa?x?y222,z?y??ya?x?y222, x2y2a1?z??z??1?2??.

222a?x2?y2a2?x2?y2a?x?y1a于是 ??d????a2?x2?y2?dxdy???adxdy?a??a2?h2?.

za2?x2?y2DxySDxy47.解：S是x?y?z?a(x?0,y?0)分解为两部分:

2222S1:x2?y2?z2?a2?x?0,y?0,z?0?, S2:x2?y2?z2?a2?x?0,y?0,z?0?.

故

??zdxdy???zdxdy???zdxdy

SS1S2???a2?x2?y2dxdy????a2?x2?y2dxdy

DxyDxy???2??a?x?ydxdy??d??ra2?r2dr?222Dxy2 0 ? a 013a?. 6?2?1?y2?2y??2?y2??y2???22??3y?f??3zdxdydz 48.解：原式= ????3x??f??????????2???V??z?z?z??y?z??z??? ?3???(x?y?z)dxdydz ?3?V4 0 222 2? 0d??4d??r2r2sin?dr

0 a ? b?2?b4?a4 ?6??sin?d??rdr?6??1????5. a2?? z b4?

o x

49.解:(Ⅰ).画出积分区域

z

21

y

o y

x

(Ⅱ).原式=

????xV2?y?z?dxdydz??22S 2? 0d??d??r2.r2sin?dr? 0 0V ? a4?5a. 522250.解：由Gauss公式，得I?xy2dydz?yz2dzdx?zx2dxdy??????(x?y?z)dV，由广

?x?arsin?cos??2义球坐标变换 ?y?brsin?sin?, J(r,?,?)?abcrsin?，得

?z?crcos??2?I??d??d??(a2r2sin2?cos2??b2r2sin2?sin2??c2r2cos2?)abcr2sin?dr

000?1?abc2??d??(a2sin2?cos2??b2sin2?sin2??c2cos2?)sin?d??050

4??abc(a2?b2?c2).15

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