?202x2x?2x?2dx;??/20sinxsinx?cosxdx?a0dxa0证(1)I?a0?a0f(x)f(x)?f(a?x)f(x)?f(a?u)f(u)?f(a?u)f(a?u)dua0du2I?a0?f(x)?f(a?x)f(x)dx?a0?f(u)?f(a?u)f(a-x)??f(x)?f(a?x)202dx??f(x)?f(a?x)202dx??1dx?a,I?a2
.解(2)?x2x?2x?2sinxdx?2?x2x?(2?x)dx?2?222?2.??/20sinx?sin(?/2?x)dx??/22??4.21.设f(x)?解f(x)?df(x)dx?tanxsinx(1?xt)dtdx求22df(x)dx.tanxsinx2?tanxsinx2(1?xt)dt?tanx?sinx?x?22tdt,2?secx?cosx?xtanxsecx?xsinxcosx?tanx?tanxsinxtdt23t?2222?secx?cosx?xtanxsecx?xsinxcosx???3?sinx
??secx?cosx?xtanxsecx?xsinxcosx??secx(1?xtanx)?cosx(1?xsinx)?22.求定积分I?解I?222222213?tan3x?sinx?313?tan3x?sinx?.3??/20cos3?d?的值.12??/20cos3?d??2?2?/20(1?cos6?)d???4?112sin6?|0?/2??4.23.求定积分I???2?0|sinx?cosx|dx的值.解I=2?|sinx?cosx|dx0?2???/20|sinx?cosx|dx??????/2|sinx?cosx|dx|sin(????t)|dx?2???/2=2??|sinx?cosx|dx?0??/20?2?t)?cos(???2???2?/20|sinx?cosx|dx?(cosx?sinx)dx??/4?/20|cost?sint|dx???/20?/40???/2/4(sinx?cosx)dx??/2(cost?sint)dx?/2?2.?2??sinx?cosx?|0?(?cosx?sinx)|?/4?(sinx?cosx)|0??4
24.设0?x0?x1,求定积分I?解I???x1x0(x?x0)(x1?x)dx的值.?x1x02(x?x0)(x1?x)dx????x1x0?x?(x1?x0)x?x0x1dxx?x0?(x1?x0)???x?1??x0x1dx?24??x?x0?(x1?x0)x1?x0?????x?1?dxu?x????242??????u??222222?x1x0?x1x0?(x1?x0)/2?(x1?x0)/2a0(x1?x0)4x1?x022du?2???u???a?udx(a?2)aa?u222u?2?aarcsina??20?a2??(x1?x0)8
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25.求下列曲线所围图形的面积:2(1)y?x?6x?8与y?2x?7.?y?x?6x?82解?2x?7?x?6x?8,?y?2x?7x?8x?15?0,(x?3)(x?5)?0.122 x
?3,x2?5.S??53(2x?7?(x?6x?8))dx?352?53(?x?8x?15)dx2?x?2????4x?15x? ?3?43?343.(2)y?x?x?16x?4与y?x?6x?8x?4.42?y?x?x?16x?43?232解?x?16x?4?6x?8x?4,x?6x?8x?0,42??y?x?6x?8x?4x?0,x?6x?8?0,(x?2)(x?4)?0,x?2,4.S?4243
?20{(x?x?16x?4)?(x?6x?8x?4)]dx42434342??[(x?6x?8x?4)?(x?x?16x?4)]dx2
2??(x3?6x24?8x)dx?3x2?8x)dx0?(?x?62 42??x?44??2x3?4x2??x ?4??042x3?4x2?????8.???2
(3)y2?x?1与y?x?3.
解(x?3)2?x?1,x2?7x?10?0,
(x?2)(x?5)?0,
x?2,5.y??1,2. 2S? ?2?1[(y?3)?(1?y)]dx 32 ??yy2?9????32?2y??.??12
(4)y?sinx,y?cosx与x??/2.解S??/2??(sinx-cosx)dx?(?cosx?sinx)|?/2/4?/4?2?1; S??5/4?5?/4?(sinx?cosx)dx?(?cosx?sinx)|/2?/2?2?1. 5? 4 ?? 4226.设区域?由曲线y?cosx,y?1及x??/2所围成,将?绕x轴旋转一周,得一旋转体V.试用两种不同的积分表示体积V,并且求V的值. 1解V=??/2?(1-cos21x)dx?20??y???2?arccosy?y?20?d?????yarcsinydy0/22V???2?0(1?cos2x)dx??4.11V?2??0yarcsinydy????0arcsinydy2??arcsiny(y)2210???10y?211?y2dx??2???y?2?1?y?212arcsiny???1
??202??24??24.27.求下列定积分的值:(1)?(2)?22duuu?1(91x212??22duu21?1/u972??1/2dx1?x21/2?arcsinx|1/2?1/2?4??6??12.200?200?80x33?5580x?1)dx?400.2028.设f(x)在[0,7]上可积,且一直已知?(1)求(2)求f(x)dx?5,?f(x)dx?6,?2570f(x)dx?3.?5075f(x)dx的值;f(x)dx的值.?
(3)证明:在(5,7)内至少存在一点,使f(x)?0.解(1)?f(x)dx?05?720f(x)dx??552f(x)dx?5?6?11.(2)?75f(x)dx??0f(x)dx??0f(x)dx?3?11??8.证(3)若不然,f(x)?0,x?(5,7),?75f(x)dx?0,但是?75f(x)dx??8?0,矛盾.29.设f(x)?sinx,h(x)?(1)??/2??/23?1, ???x?2,,g(x)?试求下列定积分的值或表达式:?2x?2, 2?x??.1xf(x)g(x)dx;(2)?g(x)h(x)dx;(3)?1?/2f(t)g(t)dx.解(1)?31?/2??/2f(x)g(x)dx???/2??/2sinxdx?0.(2)?g(x)h(x)dx???21g(x)h(x)dx?1x2?5632g(x)h(x)dx
?21xx1dx?2?322xdx??2x?12x3?2.(3)??/2?sintdt??cosx,t???x?2???/2f(t)g(t)dx??2x?sintdt??22sintdt?cos2?2cosx,2?x??.???/230设函数f(x)在区间[a,b]上连续，严格单调递增(a?0),g(y)是f(x)的反函数,利用定积分的几何意义证明下列公式?baf(x)dx?bf(b)?af(a)??f(b)f(a)g(y)dx.
并作图解释这一公式.解
b31.(1)设函数?(x)在[0,??)上连续且严格单调递增,又设当x?+?时?(x)???且?(0)=0.证明:对于任意实数a?0,B?0,下列不等式成立:aB??a0?(x)dx?(x)是a?B0??1(x)dx其中??1?a0?(x)的反函数.证由30题,??(x)dx?0??(a)0??1(x)dx?a?(a)(*).B?0时不等式显然成立.设B?0??(0),由于x?+?时?(x)???,存在a??0,?(a?)?B,?在[0,a?]连续,根据连续函数的中间值定理,存在a1?0,?(a1)?B.若a1?a,则由(*)得aB?若a1?a,则??(x)dx?0a?Ba0?(x)dx??1?B0??1(x)dx.?0?(x)dx??a0?(x)dx??B?(a)0??1(x)dx???B(a)??1(x)dx?a?(a)????a?(a)???(a)??1(x)dx?1a(?(a))(B??(a))?aB.若a1?a,则??(x)dx?0?B0??1(x)dx??a0?(x)dx???(a)0??1(x)dx???(a)B??1(x)dx?a?(a)???(a)B?1??1(x)dx?a?(a)??(?(a))(?(a)?B)?aB.1p?1q?1,证明下列Minkowski
(2)利用(1)中的不等式,对于任意实数a,b?0,p,q?1ap不等式ab?p?bqq.p?1证不妨设p?1.在(1)中取?(x)?xab?,则??1(x)?xap1/(p?1).?ap?a0xdx?p?b0x1/pdx?app?b1/(p?1)?11/(p?1)?1xex2?p?bp/(p?1)p/(p?1)p?bqq.32.设a?0,求a的值,使由曲线y?1?旋转所得之旋转体的体积等于2?.,y?1及x?a所围成的区域绕直线y?1
解??a0(y-1)dx?2?.?(022axe2x2)dx?2,12?a0xe2xdx?2,1?4a0e2x2d2x?2,?42a02edu?2,eu2a2
?1?8,2a2?ln9,a?ln3.33.作由极坐标方程r?1?sin2?所确定的函数的图形,并求它所围区域的面积.解S???0(1?sin2?)d??2??0(1?2sin2??1?cos4?2)d??3?2.