北大版高等數(shù)學(xué)課后答案7.pdf

? ?7.1 3.?f(x,y)?D?, g(x,y)?D?,?g(x,y)?f(x,y)g(x,y)?D? ?.?:?D?(x0,y0)RR D f(x,y)g(x,y)d = f(x0,y0) RR D g(x,y)d. ?.?m,Mf?D?,?.?mg(x,y) f(x,y)g(x,y) Mg(x,y). ?RR D mg(x,y)d RR D f(x,y)g(x,y)d RR D Mg(x,y)d. ?RR D g(x,y)d = 0,?RR D f(x,y)g(x,y)d = 0,?(x0,y0) D? ?.?m RR D f(x,y)g(x,y)d RR D g(x,y)d M.?,?(x0,y0) D? f(x0,y0) = RR D f(x,y)g(x,y)d RR D g(x,y)d ,?RR D f(x,y)g(x,y)d = f(x0,y0) RR D g(x,y)d. 4.?f(x,y)?D?,?,?RR D f(x,y)dxdy = 0.? ?f(x,y) = 0,?(x,y) D?. ?.?f?,?f?,?f?P D?0.?f?,? ?P?f?1 2f(P). ?RR D f(x,y)dxdy 0,?. ?7.2 ?. 3. RR D ydxdy,?D?y = 0?y = sinx (0 x )?. I = R 0 dx Rsinx 0 dyy = R 0 dxsin 2x 2 = 4. 4. RR D xy2dxdy,?D?x = 1, y2= 4x?. I = R2 2dy R1 y2/4dxxy 2 = R2 2dy 1 2(1 y4 16)y 2 = 32 21. 5. RR D e x ydxdy, ?D?y2= x, x = 0, y = 1?. I = R1 0 dy Ry2 0 dxe x y = R1 0 dyyey= 1. 6. R1 0 dy R1 y 1 3 1 x4dx =R1 0 dx Rx3 0 dy1 x4= R1 0 dxx31 x4= 1 6. 7. RR D (x2+ y)dxdy,?D?y = x2, x = y2?. I = R1 0 dx Rx x2 dy(x2+ y) = R1 0 dx(1 2x + x 5 2 3 2x 4) =33 140. 8. R 0 dx R x siny y dy = R 0 dy Ry 0 dxsiny y = R 0 dy siny = 2. 9. R2 0 dx R2 x 2ysin(xy)dy = R2 0 dy R2 0 dx2ysin(xy) = R2 0 dy2(1cos2y) = 4sin4. 1 課后答案網(wǎng) 課后答案網(wǎng) 10. RR D y21 x2dxdy, D = (x,y) | x2+ y2 1. I = 4 R1 0 dx R 1x2 0 dyy21 x2= 4 R1 0 dx1 3(1 x 2)2 = 32 45. 11. RR D (|x| + y)dxdy, D = (x,y) | |x| + |y| 1. I = RR D |x|dxdy + RR D ydxdy = 4 R1 0 dx R1x 0 dyx + 0 = 4 R1 0 dxx(1 x) = 2 3. 12. RR D (x + y)dxdy,?D?x2+ y2= 1, x2+ y2= 2y?. I = RR D xdxdy + RR D ydxdy = 0 + 2 R 3 2 0 dx R 1x2 11x2 ydy = 2 R 3 2 0 dx(1 x2 1 2) = 3 3 4 . ?. 13. R1 0 dx R 1x2 0 (x2+ y2)dy = R 2 0 d R1 0 r2rdr = 8. 14. R0 1dx R0 1x2 2 1+x2+y2 dy = R3 2 d R1 0 2 1+rrdr = (1 ln2). 15. R2 0 dx R1(x1)2 0 3xydy = R 2 0 d R2 cos 0 3rcosr sinrdr = R 2 0 d12cos5sin = 2. 16. RR 0 dx RR2x2 0 ln(1+x2+y2)dy = R 2 0 d RR 0 ln(1+r2)rdr = 4(1+R 2)ln(1+ R2) R2. 17. RR D 1 x2dxdy, D ?y = x, y = x ( 2 0), x2+ y2= a2, x2+ y2= b2(b a 0)?. I = Rarctan arctan d Rb a 1 (r cos)2rdr = ( )ln b a. 18. RR D rd,?D?r = a(1 + cos)?r = a (a 0)? ?. I = R 2 2 d Ra(1+cos) a rrdr = R 2 2 da3(cos + cos2 + 1 3 cos3) = (22 9 + 2)a 3. 19.?:? = , r = ?r = r() ( )?D?1 2 R r() 2d. ?. S = RR D dxdy = R d Rr() 0 rdr = 1 2 R r() 2d. 20.?r = a(1 + cos) (a 0,0 0,?I(a) = Ra 0 ex 2dx, J(a) =RR Da ex 2y2dxdy, ?Da= (x,y) | x2+ y2 a2,x 0,y 0.? (1) I(a)2= RR Ra ex 2y2dxdy, ?Ra= (x,y) | 0 x a,0 y a; (2) J(a) I(a)2 J(2a); (3)?10?lim a+ Ra 0 ex 2dx = 2 . ?. (1) I(a)2= Ra 0 ex 2dxRa 0 ey 2dy =RR Ra ex 2y2dxdy. (2) Da Ra D2a,?RR Da ex 2y2dxdy RR Ra ex 2y2dxdy RR D2a ex 2y2dxdy. (3) J(a) = R 2 0 d Ra 0 er 2rdr = 4(1e a2). ?lim a+ J(a) =lim a+ J(2a) = 4. ?,?lim a+ I(a) = 2 . ?7.3 ?. 1. RRR (z + z2)dV ,?x2+ y2+ z2 1. I = RRR zdV + RRR z2dV = 0 + R2 0 d R 0 d R1 0 drr2sin(rcos)2= 4 15. 3 課后答案網(wǎng) 課后答案網(wǎng) 2. RRR x2y2zdV ,?2z = x2+ y2, z = 2?. I = R2 0 d R2 0 rdr R2 r2 2 dz(rcos)2(rsin)2z = R2 0 (sin2cos2)d R2 0 r5(2 r4 8 )dr = 4 128 15 = 32 15 . 3. RRR x2sinxdxdydz,?z = 0, y + z = 1?y = x2? ?. ?Oyz?,?x?,?0. 4. RRR zdxdydz,?x2+ y2= 4, z = x2+ y2?z = 0?. I = R2 0 d R2 0 rdr Rr2 0 dzz = 2 16 3 = 32 3 . 5. RRR (x2 y2 z2)dV , : x2+ y2+ z2 a2. RRR z2dV= R2 0 d R 0 d Ra 0 r2sindr(r cos)2= 4 15a 5. ?RRR x2dV= RRR y2dV = 4 15a 5. ?I = 4 15a 5. 6. RRR (x2+ y2)dV , : 3px2+ y2 z 3. I = R2 0 d R1 0 rdr R3 3rdzr 2 = 2 R1 0 3(1 r)r3dr = 3 10. 7. RRR (y2+ z2)dV , : 0 a2 x2+ y2+ z2 b2. ?x = rcos, y = rsincos, z = rsinsin. ?I = R2 0 d R 0 d Rb a r2sindr(r sin)2= 2 4 3 b5a5 5 = 8 15(b 5 a5). 8. RRR (x2+ z2)dV , : x2+ y2 z 1. I = R2 0 d R1 0 rdr R1 r2 dz(r2cos2 + z2) = 12 + 4 = 3. 9. RRR z2dV , : x2+ y2+ z2 R2,x2+ y2 Rx. I = 4 R 2 0 d RR cos 0 2rdr RR2r2 0 dzz2= 4 R 2 0 d 1 15R 5(1 sin5) =2 15( 16 15)R 5. 10. RRR (1+xy +yz +zx)dV ,?x2+y2= 2z?x2+y2+z2= 8? ?z 0?. ?I = RRR 1dV = R2 0 d R2 0 rdr R8r2 r2 2 dz = 216 214 3 . 11. RRR (x2+ y2)dV , ?z = pR2 x2 y2?z = px2 + y2?. I = R2 0 d R 4 0 d RR 0 r2sindrr2sin2 = 2( 2 15 2 12)R 5. 12. RRR px2 + y2+ z2dV , ?z = x2+ y2+ z2= z?. 4 課后答案網(wǎng) 課后答案網(wǎng) I = R2 0 d R 2 0 d Rcos 0 r2sindrr = 10. 13. RRR z2dV , : p3(x2 + y2) z p1 x2 y2. I = R2 0 d R 6 0 d R1 0 r2sindrr2cos2 = 2 15(1 33 8 ). 14. RRR zdV x2+y2+z2 , ?x2+ y2+ z2= 2az?. I = R2 0 d R 2 0 d R2a cos 0 r2sindr cos = 16 15 . 15. RRR 2xy+1 x2+y2+z2dV , ?x2+ y2+ z2= 2a2?az = x2+ y2?z 0?. ?I = RRR 1 x2+y2+z2dV = R2 0 d R 4 0 d R 2a 0 r2sindr 1 r2 + R2 0 d R 2 4 d R a cos sin2 0 r2sindr 1 r2 = 2a(2 1 + ln 2). 16. RRR dV x2+y2+(z2)2 , : x2+ y2+ z2 1. I = R2 0 d R1 0 dr R 0 dr2sin 1 r24r cos+4 = 2 R1 0 drr2 |r+2|r2| r = 2 3. 17. RRR (x3+ siny + z)dV , ?x2+ y2+ z2 2az, px2 + y2 z?. ?I = RRR zdV = R2 0 d R 4 0 d R2a cos 0 r2sindrr cos = 7 6a 4. 18. RRR (x2y + 3xyz)dV , : 1 x 2, 0 xy 2, 0 z 1. ?u = x,v = xy,w = z.?x = u,y = v u,z = w, D(x,y,z) D(u,v,w) = 1 u. I = R2 1 du R2 0 dv R1 0 dw(uv + 3vw)1 ududvdw = 2 + 3ln2. 19. RRR (x + 1)(y + 1)dV , : x2 a2 + y2 b2 + z2 c2 1. ?I = RRR (xy+x+y+1)dV = RRR 1dV = R2 0 d R 0 d R1 0 abcr2sindr = 4 3abc. 20. RRR (x + y + z)dV , : (x x0)2+ (y y0)2+ (z z0)2 a2. ?x = x0+ u, y = y0+ v, z = z0+ w, : u2+ v2+ w2 a2.? D(x,y,z) D(u,v,w) = 1. I = RRR (x0+ y0+ z0+ u + v + w)dudvdw = RRR (x0+ y0+ z0)dudvdw = 4 3a 3(x0 + y0+ z0) 21.?,?I = RRR f(px2+ y2+ z2)dV? ?,?x2+ y2+ z2 z?z = p3x2 + 3y2?. ?.?: I = R2 0 d R 3 4 0 rdr R1+ 14r2 2 3x2+3y2 f(r2+ z2)dz. 5 課后答案網(wǎng) 課后答案網(wǎng) ?: I = R2 0 d R 6 0 d Rcos 0 f(r)r2sindr. 22.?I = Ra 0 dx Rx 0 dy Ry 0 dzf(z)dz?. ?.?0 z y x a.?I = RRR f(z)dxdydz = Ra 0 dz Ra z dy Ra y dxf(z) = 1 2 Ra 0 f(z)(z a)2dz. ?7.4 1.?z = p3a2 x2 y2?x2+ y2= 2az? ?(a 0). ?. S = RR x2+y22a2 ( q 3a2 3a2x2y2 + q 1 + x2 a2 + y2 a2)dxdy = R2 0 d R2a 0 rdr( 3a 3a2r2+ q 1 + r2 a2) = 16 3 a2. 2.