南京邮电大学 《高等数学》(下册) 习题参考答案
第七章 习题7.1 2.(1) (3)
??(x?y)D?12d????(x?y)3d?; (2)
D2??(x?y)D3d????(x?y)2d?;
D???xyzdv????xyzdv; (4)???(x???y2?z2)2dv????(x2?y2?z2)dv;
?33. (1)0?I??2; (2)36??I?100?; (3)?32??I?323?;
33习题7.2
1.(1)
?r?r40dx?2x0f(x,y)dyf(x,y)dy或或?40dy?y2f(x,y)dx;
4?rr2?y2y(2)
?dx?r2?x20?dy?0?r2?y2f(x,y)dx;
22(3)
?121dx?1f(x,y)dyx4?x21x或?1?x24?x2?1dy?1f(x,y)dx??dy?f(x,y)dx;
2y1y12(4)??1dx?1?x2f(x,y)dy???1dx??24?y2?1f(x,y)dy???1?2dx?4?x2?4?x2f(x,y)dy??dx?124?x2?4?x2f(x,y)dy或
?1dy??4?y2f(x,y)dx??dy??24?y2?4?y2f(x,y)dx??1?1dy??1?y2?4?y2f(x,y)dx??1?1dy?4?y2.1?y2f(x,y)dx.
2.(1)
?10dx?f(x,y)dy; (2)
x1?40dx?xf(x,y)dy;
2x (3)
?dx??111?x20ef(x,y)dy; (4)
?1?1dy?y?12y?12f(x,y)dx;
1 (5) 3.(1)
?10dy?yf(x,y)dx; (6)
e?0?1dy???2arcsinyf(x,y)dx??dy?0??arcsinyarcsinyf(x,y)dx.
203?69?; (2)?; (3); (4)e?e?1; (5); (6)?1. 342255?717. 5. . 6. . 4. 3629.(1)
?2?0d??f(?cos?,?sin?)?d?; (2)
ab???d??2?22cos?0f(?cos?,?sin?)?d?;
(3) 10.(1)
?1?20d??(cos??sin?)?10f(?cos?,?sin?)?d?.
?csc???40d??sec?0f(?cos?,?sin?)?d????2d??40f(?cos?,?sin?)?d?;
1
? (2)
??d??342sec??0f(?)?d?; (3)
?20d??1(cos??sin?)?1f(?cos?,?sin?)?d?;
? (4)
?40d??sec?sec?tan?f(?cos?,?sin?)?d?.
3?4a; (2) 11.(1) 412.(1) 2?3?2. 2?1; (3) ?(e?1); (4)
644?22; (2)
?8(??2); (3) 14a4; (4)
2?(b3?a3). 313.
?4a2.
2?. 37e?11; (3) ?ab. 15. (1) ln2; (2)
32214.(1)6?; (2) 16.(1)提示：作变换?习题7.3 1.(1) (3)
?u?x?y?u?x; (2)提示：作变换?.
v?y?xv?x?y??1?dx??111?x2?1?x2dy?2x2?y2x?y2f(x,y,z)dz; (2)
?dx??111?x2?1?x2dy?2xy02?x2x?2y2f(x,y,z)dz;
?dx??111x2dy?0f(x,y,z)dz; (4)
?dx?011?x0dy?f(x,y,z)dz.
115?; (2) (ln2?); (3) 0; (4) h2R2; (5) ?2?. 3642841716?; (3) ?. 4. (1) ; (2)
38124744?(A5?a5). 5. (1) ?; (2) ?a; (3)
61552. (1)
6.直角坐标系 柱面坐标系 球面坐标系 7.(1)
?dx??111?x2?1?x21dy??2?x2?y2x2?y2f(x,y,z)dz;
??2?02?d??d??02??2f(?cos?,?sin?,z)?dz;
?0d??4d??020f(rsin?cos?,rsin?sin?,rcos?)r2sin?dr.
32?3?2; (2) ?a3; (3) ; (4) ?(55?4).
26332248. 4?tf(t). 9.k?R. 习题7.4 1. (
551?2?)?a2. 2. 662?. 3. 16R2.
2
4bb2?ab?a2; (2)x?5.(1)x?0,y?,y?0; 3?2(a?b)?3?3(A4?a4)??? (3)? (4) 0,0,0,0,;??. 33??4?8(A?a)???7296,Iy?. 5784721126a; (3) a?. 7. (1) a; (2) x?0,y?0,z?154536.Ix?8. Fx?Fy?0,Fz??2?G?[(h?a)?R?a?R?h]. 总习题7
1.(1) (C); (2) (A); (3) (B); (4) (D); (5) (B),(D).
22222?4; (2) 0; (3) 2?; (4) 4?; (5) ?R4. 331423.(1) ?R?9?R; (2)?.
42. (1) 4. (1)
25016?823?; (2)?a. 33h3?hf(0)]. 5.?[3第八章 习题8.1 1.(1)Ix??LLLy2?(x,y)ds,Iy??x2?(x,y)ds;
Lx?(x,y)ds? (2)x?,??(x,y)ds2. (1) 2?a2n?1y?(x,y)ds?y?. ??(x,y)dsLL; (2)
1?(55?62?1); (3) ea(2?a)?2; 124 (4)
25633a. (1?e?2); (5) 9; (6) 1523.质心在扇形的对称轴上且与圆心的距离为
asin??处. 4.6k?.
141k3?3?a2?; (5) 13; (6) . 6. (1) ?a; (2) ?2?; (3) ?; (4)
21523?37. (1)
3432; (2) 11; (3) 14; (4) . 333
8. mg(z2?z1); 9. 10. (1) (3)
?3a. 2P(x,y)?2xQ(x,y)1?4x2?P(x,y)?Q(x,y)2Lds; (2)
?Lds;
?[L2x?x2P(x,y)?(1?x)Q(x,y)]ds.
11.
?P(x,y,z)?2xQ(x,y,z)?3yR(x,y,z)1?4x?9y22Lds.
习题8.2
1. (1) 8; (2)
1. 302. (1) 12; (2) 0; (3) 3. (1)
?2a; (4)
4?24; (2)
sin27?. 4653; (2) 236; (3) 5; (4) ?. 22121222322yy4. (1) x?2xy?y; (2) ycosx?xcosy; (3) xy?4xy?12e?12ye.
22习题8.3 1. Ix?3. (1)
22(y?z)?(x,y,z)dS. ???13149111?; (3) ?. ?; (2) 301032764; (3) ?a(a2?h2); (4) 2a4. 4. (1) 461; (2) ?4152?(63?1). 5. 153112?R7; (2) ?; (3) ; (4) . 6. (1)
2105287.(1)
3223(P?Q?R)dS; (2) ??555????2xP?2yQ?R1?4x?4y22dS.
8. 8?.
习题8.4 1. (1)
1223; (2) ?a5; (3) 81?; (4) ?a5; (5) 4?.
5523a2); (3) 108?. 2. (1) 0; (2) a(2?6xy23. (1) 2x?2y?2z; (2) ye?xsin(xy)?2xzsin(xz); (3) 2x.
习题8.5
4
1. (1) ?3?a2; (2) ?2?a(a?b); (3) ?20?; (4) ?2. (1) 2i?4j?6k; (2) i?j;
9. 2 (3) [xsin(cosz)?xy2cos(xz)]i?ysin(cosz)j?[y2zcos(xz)?x2cosy]k 3. (1) 0; (2) ?4. 4. (1) 2?; (2) 12?; 6. 0. 总习题8
1. (1) 12a; (2) 4?a; (3) 4; (4) ?6?;
(5) 2?R3(?2??2??2); (6) 2?R3; (7) (C); (8) (B). 2.(1)2ln3?(1??4)ln2??2?22?2?2arctan2;
(2) 18?; (3) 0; (4) ?a2; (5) 3. (1) 2?arctan2?. 161H; (2) ??h4; (3) 2?;
4R14. 8. 5. . 6. 2.
27.??8.
a3,??b3,??c3,Wmax?3abc. 93. 2习题9. 1
n2n?1 1. (1) un?; (2) un?(?1)n?1;
(n?1)ln(n?1)nxn?1sinnx(3)un?; (4) un?(?1)n?1.
(n?1)!n2. (1) 收敛; (2) 发散; (3) 收敛; (4) 发散.
3. (1) 发散; (2) 发散; (3) 发散; (4) 收敛; (5) 收敛; (6) 发散. 4. 提示：利用数列收敛与其子列收敛之间的关系. 5. 提示：s2n?1?s2n?u2n?1.
习题9. 2
1. (1) 发散; (2) 收敛; (3) 发散; (4) 收敛; (5) 收敛; (6) 收敛. 2. (1) 发散; (2) 收敛; (3) 发散; (4) 收敛; (5) 收敛; (6) 收敛；
(7) 收敛; (8)b?a时收敛，b?a时发散，b?a不能确定.
5