天津科技大学高等数学试题库定积分答案.docx
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天津科技大学高等数学试题库定积分答案
定积分
一、填空题
难度系数0.2以下:
12的值是.1.由定积分的几何意义可知,定积分1,xdx,/4,0
a2222.由定积分的几何意义知_________.axx,,da,/2,,a
23.由定积分的几何意义知________.xxd,3/2,,1
4.由定积分的几何意义知__0______.sindxx,,,,
25.一物体以速度做直线运动,则物体在到这段时间内行vttms,,3()t,0t,3
进的路程为________.45/2
11236.比较大小,_______.(用“”、“”或“”填空),,,,xxdxxd,,00
2237.比较大小,_______.(用“”、“”或“”填空),,,,xxdxxd,,11
,3228.比较大小,__,__.(用“,”、“,”或“”填空),sindxxsindxx,,00
5529.比较大小,__,___.(用“,”、“,”或“,”填空)lndxx(ln)dxx,,33
1d210.0.sindxx,,0dx
d2211..sinxsindxx,,dx
xd2212..tt,sinxsind,0xd
0d2213..xx,,sinxsind,xxd2xd2414..tt,2xsinxsind,0xd
x22,t-x15._________________________.dedt,edx,,,0
1sintsinx,,16.ddt,_________________________.,dx,,,xtx,,
xsinx,,217._________________________.dxdsindtt,,,,0,,2x
x2tedt,118.求极限____________________.e,lim,1xlnx
x2sindtt1,019.求极限____________________.,lim3,0x3x
x2arctandtt,0120.求极限.,lim33,0xx
a121.若,则的值等于________2____________.xx,,a(2+)d3ln2,1x
a22.若,则________-2____________.a,(21)d4xx,,,,a
2223.已知,则_______9__________.fxx()d3,[()+3]dfxx,,,00
222224.由不等式所确定区域的面积A,.xya,,,a
22xy25.由椭圆所围成图形的面积A,.,,1,ab22ab
122226.由圆与直线所围成图形的面积A,.yax,,y,0,a2
1227.由圆与直线所围成图形的面积A,.xy,,1,x,02
28.由曲线,,与直线所围成图形的面积A,2.y,2yx,x,0
29.由曲线A,yx,sin与直线,xx,,0,,所围成图形的面积2.y,0
30.由曲线与直线,所围成图形的面积A,1.yx,cosy,0xx,,0,2
22A,31.由不等式所确定区域的面积.14,,,xy3,
难度系数0.2—0.4:
xe,,x21.dlndxt,_________________________.x(e,2lnx),,2,x,,
lnx2.设为上的连续函数,且,fx()[1,),,Fxftt()()d,,1
1,则________.Fx(),Fxfx()(ln),x
x2(3sin)dttt,,013.求极限.lim,22,0x3x
sinx2,tetd,04.求极限______1____________.,lim,x0x
1,,1x5..ex,ed2,1x
1xx,6.0.xeex()d,,,,1
325xxsin7.0.,dx24,,5,,1xx
5x,18..dx,42arctan2,,1x
2x,219.设连续,且,则.fx()f
(2),fttx()d3,,,41
2x1,,tt1,10.若,则.f
(1),fxt()d,,2301,,tt
4311..,,(1sin)d,,,,,03
aa12.若,则.2bxxxbasind(0),,xxxx(sincos)d,,,,,0a
x,xA,13.由曲线,,与直线所围成图形的面积y,ey,ex,11.e,,2e
,,A,14.由曲线,在0,上所围图形的面积yx,sinyx,cos,,4,,
.2,1
2A,15.用定积分表示由曲线与直线及所围成图形的面积y,x,4x,1x,3
4.
22216.由圆所围图形绕轴旋转一周形成一个球体,其体积值=xya,,xV43.,a3
难度系数0.4—0.6:
,,11.反常积分x,当取时收敛.dk,1kk,2xx(ln)
a32222..2a()dxaxx,,,,,a
xt3.函数在上的最大值是2.[0,1]fxet()d,,0
224.由单位圆所围图形绕轴旋转一周形成一个球体,其体积值=xy,,1yV4.,3
5.用定积分表示曲线方程上对应一段弧长的弧长的值=yx,lns38,,x
13.1ln,22
难度系数0.6以上:
xelnt,1.若,则1.xfxx()d,fxt()d,,,11t
xx1Fxfttt,,2.设正值函数在上连续,则函数在上fx()[,]ab()()dd(,)ab,,abft()至少有1个根.
23.一立体以抛物线与直线围成区域为底,而用垂直于轴的平面截得的yx,xx,4
截面都是正方形,则平行截面面积Sx()=;其体积=.4xV32
二、单项选择题
难度系数0.2以下:
121.定积分值的符号为(B).xxxlnd1,2
(A)大于零;(B)小于零;(C)等于零;(D)不能确定.2.下列等于1的积分是(C).
11111(A);(B);(C);(D).xxd
(1)dxx,1dxdx,,,,00002
1xx,3.(D).(+)deex,,0
121(A);(B);(C);(D).2ee,e,eee
xx224.(B).(sin+cos)dx,,022
,,(A);(B);(C);(D)0,,,1222
15.,则(C).k,(2+)d2xkx,,0
(A)0;(B)-1;(C)1;(D)2.
e11x6.与的大小关系是(A).nx,mex,dd,,10x
(A);(B);(C);(D)无法确定.mn,mn,mn,
7.下列式子中,正确的是(C).
1122232(A);(B);xxxxdd,lndlndxxxx,,,,,0011
2211xx,2(C);(D).xxxxdd,exexdd,,,,,1100
8.已知自由落体运动的速度,则落体运动从到所走的路成为tt,vgt,t,00(C).
222gtgtgt2000(A);(B);(C);(D).gt0326
b9.积分中值定理,其中(B).fxxfba()d()(),,,,a
(A)是[,]ab内任一点;(B)是[,]ab内必定存在的某一点;,,
(C)是[,]ab内唯一的某一点;(D)是[,]ab的中点.,,
x10.设在连续,,则(A).fx()[,]ab,()()dxftt,,a
(A)是在上的一个原函数;,()xfx()[,]ab
(B)是的一个原函数;fx(),()x
(C)是在上唯一的原函数;,()xfx()[,]ab
(D)是在上唯一的原函数.fx(),()x[,]ab
b11.设且在连续,则(B).fx()[,]abfxx()d0,,a
A);(fx()0,
(B)必存在使;fx()0,x
(C)存在唯一的一点使;fx()0,x
(D)不一定存在点使.fx()0,x
12.函数在上连续是在上可积的(B).fx()[,]abfx()[,]ab(A)必要条件;(B)充分条件;
(C)充要条件;(D)无关条件.
13.下列各积分中能够直接应用牛顿—莱布尼茨公式的是(C).
331(A);(B);lndxxdx,,102,x
02(C)tandxx;(D).cotdxx,,,,,42
xsindtt,014.极限(C).,limx,0xttd,0
(A)-1;(B)0;(C)1;(D)2.
0d215.(B).sintdt,,xdx
22(A);(B);sinx,sinx
22(C);(D).,2sinxx,sint
b16.定积分(B).()()dxaxbx,,,,a
33()ba,()ab,(A);(B);66
333()ba,ba,(C);(D).63
a17.设函数在上的连续,则(C).fx()[,],aafxx()d,,,a
a(A);(B)0;2()dfxx,0
aa(C);(D).[()()]dfxfxx,,[()()]dfxfxx,,,,00
6618.已知为偶函数且,则(D).fx()fxx()d8,fxx()d,,,,06(A)0;(B)4;(C)8;(D)16.
22,x19.(D).exd,,,2
2200422,u,t,x,x2(A);(B);(C);(D).eudetd2dex2dex,,,,,2,22,2
22xy20.由椭圆所围成图形的面积A,(A).,,194
(A);(B);(C);(D).6,9,12,36,
221.由圆与直线所围成图形的面积A,(B).yx,,4y,0
(A);(B);(C);(D).,2,3,4,
2222.由圆与直线所围成图形的面积A,(A).xay,,x,0
1112222(A);(B);(C);(D).,,a,aa,a324
xA,23.由曲线与轴,直线,所围成图形的面积(B).yx,sinx,x,02
112(A);(B);(C);(D)(32
2222A,24.由不等式所确定区域的面积(,).axya,,,4
2222(A);(B);(C);(D).,a2,a3,a4,a
难度系数0.2—0.4:
lnx,1.设,其中为连续函数,则(A).fx()Fx(),Fxftdt()(),1,x
1111(A);(B);fxf(ln)(),fxf(ln)(),2xxxx
1111(C);(D).fxf(ln)(),fxf(ln)(),2xxxx2.下面命题中错误的是(A).
b(A)若在上连续,则存在;fx()(,)abfxx()d,a
(B)若在上可积,则在上必有界;fx()[,]abfx()[,]ab
(C)若在上可积,则在上必可积;fx()[,]abfx()[,]ab
(D)若在上有界,且只有有限个间断点,则在上必可积.fx()[,]abfx()[,]ab3.下列积分值为零的是(C).
,2222A)(;(B);(xxxcosdxxxcosd,,,0,2
0222(C);(D)xxxcosd.xxxsind,,,,,,22
4.下列反常积分收敛的是(B).
,,,,11(A);(B);dxdx2,,11xx
,,,,1x(C);(D).exddx,,11x
5.下列反常积分收敛的是(C).
,,,,lnx1(A);(B);xdxd,,eexxxln
,,,,11xx(C)d;(D).d2,,eexx(ln)xxln
116.(D).,dx2,,1x
(A)2;(B)-1;(C);(D)不存在.
x7.函数在上的平均值为(B).2[0,2]
333(A);(B);(C);(D).ln23ln222ln22
3,
48.定积分的值是(C).sin2dxx,0
1133(A);(B);(C);(D).,,2222
19.关于反常积分,下列结论正确的是(C).lndxx,0
(A)积分发散;(B)积分收敛于0;(C)积分收敛于-1;(D)积分收敛于1.
222210.由不等式所确定区域的面积(,).A,axya,,,2
2222(A);(B);(C);(D).(21),,a3,a,a2,a
11.由相交于点及的两条曲线,且(,)xy(,),()xyxx,yfxygx,,(),()112212
所围图形绕轴旋转一周所得的旋转体体积=(B).fxgx()()0,,xV
xx22222,,(A);(B);,fxgxx()()d,,fxgxx()()d,,,,,,,xx11
xx22222(C);(D).,[f(x),g(x)]dx,fxgxx()()d,,,,,xx11
难度系数0.4—0.6:
sinx3421.设,,当时,是的(B)fx()gx()gxxx(),,x,0fxtt()sind,,0
无穷小量.
(A)高阶;(B)同阶非等价;(C)高阶;(D)低价.
xt2.设,则fx()(A).fxtet()
(1)d,,,0
(A)有极小值;(B)有极大值;2,ee,2(C)有极大值;(D)有极小值.2,ee,2
x3.设在上连续且为奇函数,,则(B).fx()[,],aaFxftt()()d,,a
(A)是奇函数;(B)是偶函数;Fx()Fx()(C)是非奇非偶函数;(D)(A)、(B)、(C)都不对.Fx()
11,x24.(C).coslndxx,1,,1,x2
1(A)1;(B)-1;(C)0;(D).2
bdx5.广义积分的收敛发散性与的关系是(B).()ba,kk,a()xa,
A)时收敛,时发散;(B)时收敛,时发散;(k,1k,1k,1k,1(C)时收敛,时发散;(D)时收敛,时发散.k,1k,1k,1k,1
6.曲线,,,()及轴所围图形面积(,).A,yx,lnya,lnyb,lny0,,ab
abeelnblnbxy(A);(B);(C);(D).lndxxedxlndxxedyba,,,,eelnalna
7.曲线与直线、所围图形绕轴旋转一周所形成的旋转体的体积y,0yyx,x,4
(C).V,
424(A);(B);,xxd,yyd,,00
2244(C);(D).32d,,,yy16d,,,yy,,00
难度系数0.6以上:
2xtanarctandttt,,01.若,则(D).k,lim0,,ck,0xx
(A)3;(B)4;(C)5;(D)6.
12,,,,,,2.设连续,已知,应是(C).fu()nnxfxxtftt
(2)d()d,,,00
1(A)2;(B)1;(C)4;(D).2
A,3.由心形线所围成图形的面积(D).r,,22cos,
2,1222(A);(B);,,,(22cos)d(22cos)d,,,,,002
,122(C);(D).,,,(22cos)d,,,(22cos)d,,002
三、计算题
难度系数0.2以下:
11.((23)dxx,,0
112.解:
(23)d(3)4xxxx,,,,,00
2122.(()dxxx,,,1x
211152232解:
.()d[ln]ln2xxxxxx,,,,,,,1,1x2360x3.((cos)dxex,,,,
00xx,,解:
.(cos)d(sin)1xexxee,,,,,,,,,,
1424.((3x,2x,1)dx,0
132442531解:
.(321)d[]xxxxxx,,,,,,0,05315a5.((x,a)(x,a)dx,0
332aa2xaaa2233解:
(()()d()d,xaxaxxaxaa,,,,,,,,,0,,00333
916.(x(1,)dx,4x
9911228443/29
(1)d()d[2]24x,x,x,x,x,x,,,解:
(4,,44333xx,217.(dx,,3,1x
21,2,ln2解:
(dx,ln1,x,0,ln2,,3,,31,x
,8.(sin()dxx,,,33
,,,,,,解:
.sin()dsin()d()cos()0xxxxx,,,,,,,,,,,,,3333333,9.((sincos)dxxx,,0
,解:
.(sincos)d(cossin)(11)02xxxxx,,,,,,,,,,,00
310.((sinsin2)dxxx,,0
,3113解:
.,,,,,,(sinsin2)d(coscos2)xxxxx,0240,11.((1,2sinx)dx,0
,解:
((1,2sinx)dx,,,2cosx,,,2(,1,1),,,40,0
2212.(cosdxx,,,2
,,21cos211,x,,,222解:
(cosddsin2xxxxx,,,,,,,,,,,,2222,,,22,2,213.((1cos)d,,,,0
,,,1cos211,,,22解:
(1cos)dsindd(sin2),,,,,,,,,,,,,,,,00022220
π,2214((cosd,,02
πππ,,1cos,1π,22222解:
(,,cosdd,,,,(sin)|,,0,,002224
415.(sectandxxx,0
,44解:
(sectandsec21xxxx,,,0,0
3dx16.(,21/31,x
3dx,,,3arctan解:
(,x,,,,21/31/33661,x1dx217.(,021,x
1dx,,1/22解:
(,arcsinx,,0,0,02661,x1dx18.(,204,x
x1d()11,dxx2解.,,,arcsin,,20026x,4x02,1()2
2119.(dx,204,x
21x1,2arctan解:
.,dx,0,202284,x
21x20.(dx2,01,x
22111xx,,111,1解:
(dd
(1)d[arctan]1xxxxx,,,,,,,0,,,2220001114,,,xxx31221.(()dxx,,2x
3391119322解:
(()d
(2)d(2ln)lnxxxxxxx,,,,,,,,,,,24x222x2
42x1,22.dx(,31x
4222xx,1111792解:
xxx(d,(,)d,[,],,1,1,,332118xx2x82
41x23.(dx2,01,x
144111xx,,111123解:
dx,d
(1)d(arctan)xxxxxx,,,,,,2,,,220001,x113,,xx0
2,.,,43
2112,x24.(dx122,xx
(1),3
122211112,x1111,,xx解:
dx,dxdxx,,,,,()(arctan)111,,,2222221xx
(1),xxxxx
(1)1,,3333
.,,1,312
e125.x(d,1xx,(21)
eeee11211xx,,()解:
,,xx()dddd21,,,,1111xxxx(),,2121xx,21
ee.,,,,,,,ln|ln(21)|1ln3ln(21)xxe11
3dx26.22,1xx
(1),
333d11x13,3解:
.,,()dx,,,,,,arctan1x,,2222111xxxx
(1)1,,x31212527.(
(1)dxx,,1
22115562解:
(
(1)d
(1)d
(1)
(1)xxxxx,,,,,,,1,,1166
3dx28.(,42(8,2x)
33xxd1d(8,2)111173解:
(,,,,(,),2,,44322384xxx26864(8,2)(8,2)6(8,2)
1sin3/,x29.(dx,22/,x
1sin,,3/3/111113/,x解:
(xd,,sind()•,cos,,0,2/,,,22/2/,,2xxx2x
4sinx30.(dx,1x
444sinx解:
(d2sind2cos2(cos1cos2)xxxx,,,,,,,111x
1arctanx31.(dx,201,x
1211arctan1x,2解:
(,,,darctand(arctan)(arctan)xxxx,,200,1232x0
elnx32.(dx,1x
e13eeln222x22解:
(,d(ln)dln(ln)(10)xxxx,,,,,,11x3331
xln3e33.(dxx,01,e
xln3ln3e1ln3xx解:
(dd
(1)ln
(1)2ln2xee,,,,,0,,xx0011,,ee
a2x34.(xexd,0
aa2221112xxxaa解:
(xexexee,,,,dd
(1)0,,00222
3235.(x1,xdx,0
33111722223/23解:
(x1,xdx,1,xd(1,x),(1,x),(8,1),0,,003233
236.(sincosdttt,0
,,112222解:
(,,,,,sincosdcosdcoscostttttt,,00022,437.(cosxsinxdx,0
,1445,解:
(cosxsinxdx,sinxdsinx,sinx,00,,0052,1cos2,x38.(dx,02
222,,,,1cos2,x解:
dsindsindsindxxxxxxx,,,,,,,000,2
2,,,cosx,cosx,4.0,1xx39.(2dex,0
x1(2e)2e,11xx,解:
.2dex,0,0ln(2e)1,ln2
5140.(dx,113,x
2t,12解:
令,则.于是13,,xtxxtt,,,dd33
4544112224.dddxtttt,,,,,,,,122t333313,x24141.(dx,11,x
2解:
令,则x,t,dx,2tdt.于是x,t
42212d13tt,,,,2.d21d2[ln
(1)]21lnxttt,,,,,,,,1,,,,,,,111112,,tt1,x,,,,9xdx42.(,1,1x
2解:
令,则,,于是x,tx,tdx,2tdt
2933xt1,,,,dx2dt2(t1)dt,,,111,,1t1t,1x
233(,,,,tt42ln
(1),,42ln211
1x43.(dx,,1,54x
2解:
令,则,,于是x,(5,t)/45,4x,tdx,,tdt/2
323113xttt(5)11,12(dd(5)d(5)xxttt,,,,,,,,,,,13168883t54,x1
2344.(tanxdx,0
,,,,,22333解:
(3tand(sec1)dtan30xxxxx,,,,,,,,,,,000333,2245.(cotdxx,,4
,,,22222解:
(csc1)(cot