第五章 定积分及其应用
一、填空题
1．由?a,b?上连续曲线y?f?x?,直线x?a,x?b?a?b?和x轴围成的图形的面积为 db2．?sin?x2?1?dx? dxa3．设F?x???1x1?tdt,则F??x?? 104．利用定积分的几何意义求?xdx? 5．积分?1x2lnxdx值的符号是
31?6．定积分?20?sin214x?sin5x?dx值的符号是
27．积分I1??lnxdx与I2??ln2xdx的大小关系为 18．积分I1??lnxdx与I2??ln2xdx的大小关系为
33449．区间?c,d???a,b?,且f?x??0,则I1??f?x?dx与I2??f?x?dx的大小关系
acbd为
10．f?x?在?a,b?上连续,则?f?x?dx? ?f?x?dx
abba11．若在区间?a,b?上,f?x??0,则?f?x?dx 0
ab12．定积分中值定理中设f?x?在?a,b?上连续,则至少存在一点???a,b?,使得
f???? 13．设F?x???etdt,x?0,则F??x??
0x2dx2sintdt? 14．?20dx1?cost15．设F?x???3??x?sin3tdt,??x?可导,则F??x??
?
?16．limx?0x01?t2dtx?17．limx?0x0sintdtx2x0? 18．设f?x???t?t?1?dt,则f?x?的单调减少的区间是 19．函数f?x???20．设f?x???1x03tdt在区间?0,1?上的最大值是 ,最小值是
t2?t?11?x3sint3dt,则f??x??
ba21．设F?x?是连续函数f?x?在区间?a,b?上的任意一个原函数,则?f?x?dx? 22．?2x?3xdx?
01?23．?2??2cosxesinxdx?
3f??x?24．设f??x?在?1,3?上连续,则?dx?
11?f2x??25．??1?sin2xdx?
2?26．?cos2xdx?
0?e2x?127．?xdx?
0e?1128．?2?0sinxdx?
29．?e21dx?
x1?lnxx2sin3xdx? 30．??51?x4531．设f?x?在??a,a?上连续,则?sinx??f?x??f??x???dx? ?a1?x?1,x?032．设f?x???2,则?f?x?dx?
?1?x,x?0a????cosx,x??,0?1??2fx?33．设?????,计算???f?x?dx?
2?ex,x??0,1??1dx发散,则必有q 1xq1135．若广义积分?pdx收敛,则必有p
0x34．若广义积分???36．反常积分?37．?1??0xe?xdx?
211?x20dx?
38．曲线y?x2,y2?x所围成的图形的面积为 1?39．曲线y?sin2x,y?1,x?0,x?所围成的图形的面积为
22二、单项选择题
1．函数f?x??0,x??a,b?且连续,则y?f?x?,x轴,x?a与x?b围成图形的面积s?( ) A．
b?f?x?dx B．?f?x?dxaba
C．
?f?x?dx
abf?b??f?a????b?a???D．
22．I1??lnxdx,I2??ln2xdx,则I1与I2大小关系为( )
3344A．? B．? C．? D．? 3．f?x?连续,I?t?f?tx?dx,则下列结论正确的是( )
A．I是s和t的函数 B．I是s的函数 C．I是t的函数 D．I是常数
4．f?x?连续且满足f?x??f?2a?x?,a?0,c为任意正数,则
st0?c?cf?a?x?dx?( )
A．2?f?2a?x?dx B．2?f?2a?x?dx C．2?f?a?x?dx D．0
0?c0ccc5．f?x?连续,F?x???e?xxf?t?dt,则F??x??( )
A．?e?xf?e?x??f?x?D．e?xf?e?x??f?x?
B．?e?xf?e?x??f?x?
C．e?xf?e?x??f?x?
6．设I?x???sintdt,则I??x??( )
xx2A．cosx2?cosx B．
2xcosx2?cosx C．
2xsinx2?sinx
D．2xsinx2?sinx 7．当x?0时,f?x???sinx0sint2dt与g?x??x3?x4比较是( )
A．高阶无穷小 B．低阶无穷小 C．同阶但非等价无穷小 D．等价无穷小
8.f?x?,??x?在点x?0的某邻域内连续,且当x?0时,f?x?是??x?的高阶无穷小,则x?0时,?f?t?sintdt是?t??t?dt( )
00xxA．低阶无穷小 B．高阶无穷小 C．同阶但不等价无穷小 D．等价无穷小
9．f?x?为连续的奇函数,又F?x???f?t?dt,则F??x??( )
0xA．F?x? B．?F?x? C．0 D．非零常数 10．设F?x??xxf?t?dt,f连续,则limF?x??( ) ?2x?2x?2A．0 B．2 C．2f?2? D．f?2? 11．设f?x?连续,x?0,且?f?t?dt?x2?1?x?,则f?2??( )
1x2A．4 B．22?12 C．1?322 D．12?2b2 12．设f???x?在?a,b?上连续,且f??a??b,f??b??a,则?f??x?f???x?dx?( )
aA．a?b B．
11?a?b? C．a2?b2 D．?a2?b2? 22?xet2?1dt??0,x?0,且已知f?x?在x?0点连续,则必有( ) 13．若f?x???x2?x?0?a,A．a?1 B．a?2 C．a?0 D．a??1 14．设e?t,则?A．
x??1xexe?e?x0dx?( ) e?tett?tdt
?10dt
B．
?01dt1?t
C．
?e11?t21dt
D．?e1t?t?115．f?x?在给定区间连续,则?x3f?x2?dx?( )
0a1aA．?xf?x?dx20D．?xf?x?dx
0a1a2 B．xf?x?dx?02
C．2?xf?x?dx
0a216．积分?e1lnxdx的值是( ) x D．?1
e21111A．? B．2? C．2222e217．若?x041x4ff?t?dt?,则?02x?x?dx?( )
A．16 B．8 C．4 D．2 18．积分?1?1x2dx的值是( )
A． 0 B．1 C．
1 D.2 2119．曲线y?,y?x,x?2所围平面图形的面积为( )
xA．
2?21?1???x?dx?x? B．
?211??x???dxx?? C．
?212?1?2?dy??2?y?dy ???1y??D．?121???2??dx??1?2?x?dx
x??20．曲线y?ex与其过原点的切线及y轴所围平面图形的面积为( )
A．?1010?ex?ex?dx B．
?e1?lny?ylny?dy C．
??e1ex?xex?dx
D．??lny?ylny?dy
21．在区间?a,b?上f?x??0,f??x??0,f???x??0,
令s1??f?x?dx,s2?f?b??b?a?,s3?ab1f?b??f?a????b?a?,则有( ) ??2A．s1?s2?s3 B．s2?s1?s3 C．s3?s1?s2 D．s2?s3?s1 22．曲线y?cosx,?体体积等于( )
?2?x??2与x轴围成的平面图形绕x轴旋转一周而成的旋转
【2019年整理】定积分及其应用测试题
