????rdrd?D?0??2?d??rdra1???21?a28?2
?216(1?a2)(5分)[39][答案]
??Df(x,y)d????(2x?y)dxdy??D330?3?x0(2x?y)dy
1?????2x(3?x)?(3?x)2?dx02??27?2而D的面积?=9 2 ∴所求平均值=3. (6分)[40][答案]
??Df(x,y)dxdy??dx?(x?by)dy0x12015x∵??(4x7631?72x2)dx
?而D的面积
?=?dx?dy0x5x??4xdx01
?2∴所求平均值=12(4分)[41][答案] 原式=
22 3x?y?1??1?x2?y2?22x?y?1??dxdy
22????3 2?1?3(4分)[42][答案] (3分)[43][答案] (4分)[44][答案] (5分)[45][答案]
交点为(2，1)与(2，－1)
??ydy??110125?y21?y2xdx??y2(4?3y2?y4)dy ?62105(5分)[46][答案] (4分)[47][答案]
??dy?xy2?x2dx001y113ydy?031?12?(3分)[48][答案] 原式=
?R?Rydy?2R2?y222?R?yx3dx
对于
?R2?y222?R?yx3dx被积函数x3为奇函数
∴积分为零。
故原式=0. (5分)[49][答案]
xdy221x?y22?x??(?arctan)dx
2原式=1418??arctan?ln?254??dx?x22x(4分)[50][答案]
1dy21yx21??x2(x?)dx 1x9?4??xdx?122x(4分)[51][答案]
?2?xdx?0aa2?x20dy
?2?x0a2?x2dx?23a3(5分)[52][答案]
由对称性知，此积分等于D域位于第一象限中的部分D1上的积分的4倍，在第一象限|x|=x.
?4?dy?0ba22b?yb0xdx?2?b0a222(b?y)dy 2b4?a2b3(5分)[53][答案]
22x??dx?02004x2?y2dy
???x2dx8??3(5分)[54][答案]
??ln3ln2ln3dy?yexydx24??(e4y?e2y)dy
ln2?13342p(5分)[55][答案]
??????2p2pydy22?py22pxdx?12y4y(p?2)dy 2p28p825p2112x1y(6分)[56][答案]
??xdx?2dy??xdy?2dx0x0y??(x012x?x)dx??(yy?y3)dy
041?33140(6分)[57][答案]
??dy?edx1y2y2xy??(yey?ye)dy
123?e2?e2(5分)[58][答案]
??dy?010110yyxy?y2dx??32y(x?y)203110yydy
??18y2dy?6(5分)[59][答案]
1x3??dx?01?x((12x2?16x3y3)dy
233122???12x(x?x?4x(x?x)??dx0???(12x012x?8x?4x)dx515?5184x(8分)[60][答案]
?10dx?1?xx2?y2dyx?xy2x2y2??(x?y?arcsin022x???1dx)
?20xdx2?61(3分)[61][答案]
xsinx??dx?dy00x??sinxdx01
?1?cos(4分)[62][答案]
x2sinx??dx?dy00x1??xsinxdx01
?sin1?cos1(5分)[63][答案]
?0??2d??ln(1?r2)rdr02????4?4?51lnudu
(5ln5?4)(5分)[64][答案]
???d??2?24cos?2cos?r3dr??2?260cos4?d?
0?45?2(5分)[65][答案]
???r?rdrd?D???2?d???22cos?0r2dr8???2?cos3?d?3?216???2cos3?d?3016232???339(4分)[66][答案] 原式=
?
?2?0d??sinr2?rdr
?2? =π(cosπ2－cos4π2). (4分)[67][答案]
???1?r2rdrd?D?0??2d??1?r2rdr
01??6(7分)[68][答案]