北大版高等数学课后答案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)dσ = f(x0,y0) RR D g(x,y)dσ. ?.?m,MǑf?D????,???.?mg(x,y) ≤ f(x,y)g(x,y) ≤ Mg(x,y). ??RR D mg(x,y)dσ ≤ RR D f(x,y)g(x,y)dσ ≤ RR D Mg(x,y)dσ. ?RR D g(x,y)dσ = 0,?RR D f(x,y)g(x,y)dσ = 0,?????(x0,y0) ∈ D?? ??.???m ≤ RR D f(x,y)g(x,y)dσ RR D g(x,y)dσ ≤ M.?????,??(x0,y0) ∈ D? f(x0,y0) = RR D f(x,y)g(x,y)dσ RR D g(x,y)dσ ,?RR D f(x,y)g(x,y)dσ = f(x0,y0) RR D g(x,y)dσ. 4.???f(x,y)??????D???,??,?RR D f(x,y)dxdy = 0.? ?f(x,y) = 0,?(x,y) ∈ D?. ?.?Ǒf??,?f???Ǒ?,?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 ≤ π)??. I = Rπ 0 dx Rsinx 0 dyy = Rπ 0 dxsin 2x 2 = π 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 dy√1 − x4= R1 0 dxx3√1 − x4= 1 6. 7. RR D (x2+ y)dxdy,??D?y = x2, x = y2??. I = R1 0 dx R√x x2 dy(x2+ y) = R1 0 dx(1 2x + x 5 2− 3 2x 4) =33 140. 8. Rπ 0 dx Rπ x siny y dy = Rπ 0 dy Ry 0 dxsiny y = Rπ 0 dy siny = 2. 9. R2 0 dx R2 x 2ysin(xy)dy = R2 0 dy R2 0 dx2ysin(xy) = R2 0 dy2(1−cos2y) = 4−sin4. 1 课后答案网 课后答案网 10. RR D y2√1 − x2dxdy, D = {(x,y) | x2+ y2≤ 1}. I = 4 R1 0 dx R √1−x2 0 dyy2√1 − 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 R1−x 0 dyx + 0 = 4 R1 0 dxx(1 − x) = 2 3. 12. RR D (x + y)dxdy,??DǑ?x2+ y2= 1, x2+ y2= 2y?????????. I = RR D xdxdy + RR D ydxdy = 0 + 2 R √3 2 0 dx R √1−x2 1−√1−x2 ydy = 2 R √3 2 0 dx(√1 − x2− 1 2) = π 3 − √3 4 . ??????????????????. 13. R1 0 dx R √1−x2 0 (x2+ y2)dy = Rπ 2 0 dθ R1 0 r2rdr = π 8. 14. R0 −1dx R0 −√1−x2 2 1+√x2+y2 dy = R3π 2 π dθ R1 0 2 1+rrdr = π(1 − ln2). 15. R2 0 dx R√1−(x−1)2 0 3xydy = Rπ 2 0 dθ R2 cosθ 0 3rcosθr sinθrdr = Rπ 2 0 dθ12cos5θsinθ = 2. 16. RR 0 dx R√R2−x2 0 ln(1+x2+y2)dy = Rπ 2 0 dθ RR 0 ln(1+r2)rdr = π 4[(1+R 2)ln(1+ R2) − R2]. 17. RR D 1 x2dxdy, D ??y = αx, y = βx (π 2 β α 0), x2+ y2= a2, x2+ y2= b2(b a 0)???????????. I = Rarctanβ arctanα dθ Rb a 1 (r cosθ)2rdr = (β − α)ln b a. 18. RR D rdσ,??D?????r = a(1 + cosθ)???r = a (a 0)????? ??????. I = Rπ 2 −π 2 dθ Ra(1+cosθ) a rrdr = Rπ 2 −π 2 dθa3(cosθ + cos2θ + 1 3 cos3θ) = (22 9 + π 2)a 3. 19.?????????????:???θ = α, r = β???r = r(θ) (α ≤ θ ≤ β)????D???????1 2 Rβ α[r(θ) 2]dθ. ?. S = RR D dxdy = Rβ α dθ Rr(θ) 0 rdr = 1 2 Rβ α[r(θ) 2]dθ. 20.???r = a(1 + cosθ) (a 0,0 ≤ θ 0,??I(a) = Ra 0 e−x 2dx, J(a) =RR Da e−x 2−y2dxdy, ??Da= {(x,y) | x2+ y2≤ a2,x ≥ 0,y ≥ 0}.?? (1) [I(a)]2= RR Ra e−x 2−y2dxdy, ??Ra= {(x,y) | 0 ≤ x ≤ a,0 ≤ y ≤ a}; (2) J(a) ≤ [I(a)]2≤ J(√2a); (3)?????10?????lim a→+∞ Ra 0 e−x 2dx = √π 2 . ?. (1) [I(a)]2= Ra 0 e−x 2dxRa 0 e−y 2dy =RR Ra e−x 2−y2dxdy. (2) Da⊂ Ra⊂ D√2a,??RR Da e−x 2−y2dxdy ≤RR Ra e−x 2−y2dxdy ≤RR D√2a e−x 2−y2dxdy. (3) J(a) = Rπ 2 0 dθ Ra 0 e−r 2rdr = π 4(1−e −a2). ??lim a→+∞ J(a) =lim a→+∞ J(√2a) = π 4. ?????,?lim a→+∞ I(a) = √π 2 . ??????7.3 ????????. 1. RRR Ω (z + z2)dV ,??ΩǑ???x2+ y2+ z2≤ 1. I = RRR Ω zdV + RRR Ω z2dV = 0 + R2π 0 dθ Rπ 0 dϕ R1 0 drr2sinϕ(rcosϕ)2= 4π 15. 3 课后答案网 课后答案网 2. RRR Ω x2y2zdV ,??Ω??2z = x2+ y2, z = 2??????. I = R2π 0 dθ R2 0 rdr R2 r2 2 dz(rcosθ)2(rsinθ)2z = R2π 0 (sin2θcos2θ)dθ · R2 0 r5(2 − r4 8 )dr = π 4 · 128 15 = 32π 15 . 3. RRR Ω x2sinxdxdydz,??ΩǑ???z = 0, y + z = 1???y = x2???? ?. ??Ω??Oyz????,???????x????,????Ǒ0. 4. RRR Ω zdxdydz,??Ω?x2+ y2= 4, z = x2+ y2?z = 0??. I = R2π 0 dθ R2 0 rdr Rr2 0 dzz = 2π · 16 3 = 32π 3 . 5. RRR Ω (x2− y2− z2)dV , Ω : x2+ y2+ z2≤ a2. RRR Ω z2dV= R2π 0 dθ Rπ 0 dϕ Ra 0 r2sinϕdr(r cosϕ)2= 4π 15a 5. ??RRR Ω x2dV= RRR Ω y2dV = 4π 15a 5. ??I = −4π 15a 5. 6. RRR Ω (x2+ y2)dV , Ω : 3px2+ y2≤ z ≤ 3. I = R2π 0 dθ R1 0 rdr R3 3rdzr 2 = 2π R1 0 3(1 − r)r3dr = 3π 10. 7. RRR Ω (y2+ z2)dV , Ω : 0 ≤ a2≤ x2+ y2+ z2≤ b2. ?????x = rcosϕ, y = rsinϕcosθ, z = rsinϕsinθ. ?I = R2π 0 dθ Rπ 0 dϕ Rb a r2sinϕdr(r sinϕ)2= 2π · 4 3 · b5−a5 5 = 8π 15(b 5 − a5). 8. RRR Ω (x2+ z2)dV , Ω : x2+ y2≤ z ≤ 1. I = R2π 0 dθ R1 0 rdr R1 r2 dz(r2cos2θ + z2) = π 12 + π 4 = π 3. 9. RRR Ω z2dV , Ω : x2+ y2+ z2≤ R2,x2+ y2≤ Rx. I = 4 Rπ 2 0 dθ RR cosθ 0 2rdr R√R2−r2 0 dzz2= 4 Rπ 2 0 dθ 1 15R 5(1 − sin5θ) =2 15(π − 16 15)R 5. 10. RRR Ω (1+xy +yz +zx)dV ,??ΩǑ???x2+y2= 2z?x2+y2+z2= 8? ?z ≥ 0???. ????I = RRR Ω 1dV = R2π 0 dθ R2 0 rdr R√8−r2 r2 2 dz = 2π16 √2−14 3 . 11. RRR Ω (x2+ y2)dV , Ω?z = pR2 − x2− y2?z = px2 + y2??. I = R2π 0 dθ Rπ 4 0 dϕ RR 0 r2sinϕdrr2sin2ϕ = 2π( 2 15 − √2 12)R 5. 12. RRR Ω px2 + y2+ z2dV , Ω?z = x2+ y2+ z2= z??. 4 课后答案网 课后答案网 I = R2π 0 dθ Rπ 2 0 dϕ Rcosϕ 0 r2sinϕdrr = π 10. 13. RRR Ω z2dV , Ω : p3(x2 + y2) ≤ z ≤ p1 − x2 − y2. I = R2π 0 dθ Rπ 6 0 dϕ R1 0 r2sinϕdrr2cos2ϕ = 2π 15(1 − 3√3 8 ). 14. RRR Ω zdV √ x2+y2+z2 , Ω?x2+ y2+ z2= 2az??. I = R2π 0 dθ Rπ 2 0 dϕ R2a cosϕ 0 r2sinϕdr cosϕ = 16π 15 . 15. RRR Ω 2xy+1 x2+y2+z2dV , Ω Ǒ?x2+ y2+ z2= 2a2?az = x2+ y2??z。