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Файл:Mathematics for Foreign Students integrals. Study Guide
.pdf
81
From the figure 20. we note that , so the volume of the
solid is given by the following
=
Exercises
1. Find the area of the region which is defined by the function curve
, the -axis, and the lines
2. Find the area which is defined by the function curve
and the -axis.
3. Find the area which is defined by the function curve
and the -axis.
In the following exercises, find the area of the region which is defined
by function and the -axis on the specified intervals.
7. Find the area of the region which is defined by the curves of func-
tions , and the lines
8. Find the area of the region which is defined by two curves
, and the lines
9. Find the area of the region which is defined by the curves of functions
, and the lines
82
10. Find the area which is bounded by the curve
and
the line
In the following exercises, find the area which is bounded by functions
and
In the following exercises, find the areas
of regions with the fol-
lowing boundaries:
.
.
.
.
In the following exercises, find the length of arc with the following
boundaries
.
.
83
In the following exercises, find the volumes of the solids which are
generated by revolving the regions with the following boundaries:
around x-axis.
, around x-axis and around y-axis.
, around x-axis and around y-axis.
around and around .
, around
In the following exercises, find the volumes of the solids which are
generated by revolving the regions which are bounded by functions
and
, around -axis.
,
,
TEST № 1
1. Choose the correct answer for the integral
( )
1
5x
0
6x 3 2 dx− + =
1) 4–2ln3; 2)
8 2 ln3−
; 3) 8–ln3; 4) 4; 5)
3 2 ln3−
.
2. Choose the correct answer for the integral
3
dx
3x 1−
84
1)
( )
2
3
3x 1
C;
2
−
+
2)
( )
4
3
3 3x 1
С;
4
−
+
3)
( )
2
3
x1
С;
3
−
+
4)
( )
2
3
3x 1
С;
6
−
−
+
5)
( )
4
3
x3
С.
3
−
+
3. Choose the correct answer for the integral
1) 5е–2; 2) 2–3е; 3) е–2; 4) 1–3е; 5) 1.
4.
( )
( )
( )
3
2
x 3 dx
x 1 x 1
−
=
−−
1)
2x ln x 1 4 C− + − +
; 2)
35
x ln x 7 C;
x1
+ + − +
−
3)
3
x ln x 1 C
x2
− + + +
−
;
4)
2
3
x ln x 1 C
x1
− + − +
−
; 5)
3
x ln x 1 C
x1
+ + − +
−
.
5. =
1)
1 x 1 x
ln tg ln tg 3 C
5 2 3 2
+ − − +
;
2)
1 x 1 x
ln tg ln tg 3 C;
2 2 6 2
+ + − +
3)
1 x 1 x
ln tg ln tg 3 C;
5 5 3 5
+ − − +
4)
1 x x
ln tg 8 ln tg 3 C;
6 2 2
+ + + +
5)
1x
ln tgx 8 ln tg 3 C.
52
+ − − +
e
2
1
ln xdx
3cos 4sin
dx
xx+
85
6. Choose the correct answer for the integral
+
;
21
4
dx
x
x
1)
4 4 4
2
x x arctg 2 x C;
3
− + +
2)
4
3
44
x x arctg x C;− + +
3)
4
3
44
22
x x arctg 2 x C;
32
− + +
4)
4
3
44
2
x x arctg 2 x C;
3
+ + +
5)
5
44
x x 8arctg 2 x C.− + +
7. The area which is bounded by lines y=x, y=2x, x=3, is equal to:
1) 9; 2) 2; 3) 4; 4) 9/2; 5) – 9/2.
8. The length of arc of the curve which is given by the polar coordinate
8cos , =
,
34
−
is equal to:
1)
3;
2)
8
;
3
3)
11
;
3
4)
14
;
3
5)
10
.
3
9. The volume of the solid which is generated by revolving the region
around the y-axis, where the region is bounded by
x y 2=−
and y=2x+2,
is equal to:
1)
( )
4
куб.ед.
3
; 2)
( )
8
куб.ед.
3
; 3)
( )
2 куб.ед.
;
4)
( )
5 куб.ед.
; 5)
( )
куб.ед.
3
.
10. Solving the following improper integral
2
2
1
dx
I
x 4x 3
=
−+
, leads to the
next result:
1) Integral diverges; 2)
; 3)
;
4) ; 5) .
86
ТЕSТ № 2
1. Choose the correct answer for the integral
1) –12; 2) 2; 3) –3; 4) 7; 5) 11.
2. Choose the correct answer for the integral
−
dx
x
x
sin21
cos
1)
ln1 2sinx C;−+
2)
1
ln1 2sinx C;
2
++
3)
1
ln1 sinx C;
2
− + +
4)
ln1 cosx C;−+
5)
1
ln1 2sinx C.
2
− − +
3. Choose the correct answer for the integral
1) ; 2) ;
3) ; 4) ; 5) .
4. Choose the correct answer for the integral
2
2x 3
dx
x 4x 5
+
++
1) ; 2) ;
3) ;
4) ;
5)
5.
sin4xcos5xdx =
1)
11
cosx cos9x C;
2 18
++
2)
11
cosx cos9x C;
18 18
++
4
2
1
98
2
2
x x dx
x
−−
3
1
arctg xdx
2 3 3 1
ln2
12 2
−
−
4 3 3 1
ln2
12 2
−
−
12
1 2 3 3
ln2
2 12
−
−
8
2
ln 4 5x x C+ + +
2arctg( 2)хC++
2
ln 4 5 arctg( 2)xx хC+ + − + +
2
ln 4 5 2arctg( 2)xx хC+ + + + +
ln 2 arctg( 2)x хC+ − + +
87
3)
1
cosx cos9x C;
2
−+
4)
11
cosx cos9x C;
39
−+
5)
11
cosx cos9x C;
2 18
−+
6.
2
dx
x 3x 4
=
+−
1)
2
3
ln x x 4 C;
2
+ + + − +
2)
2
3 3 25
ln x x C;
2 2 4
+ + + − +
3)
2
3 25 25
ln x x C;
2 4 4
+ + + − +
4)
2
3
ln x x 7 C;
2
+ + + +
5)
1 x 2
ln C.
3
2
x
4
−
+
+
7.
2
y x 2,=+
y=0, x=-2, x=1. S-?
1) 1 ; 2) 7; 3) 9; 4) 3 ; 5) -2.
8. The length of arc which is given by the polar coordinate
,
32
−
is equal to:
1)
14 ;
2)
7;
3)
11
;
3
4)
35
;
3
5)
10
.
3
88
9. The volume of the solid which is generated by revolving the region
around the y-axis, where the region is bounded by ху=3 and у=4-х, is equal to:
1)
( )
8
куб.ед.
3
; 2)
( )
7
куб.ед.
3
; 3)
( )
2 куб.ед.
;
4)
( )
5 куб.ед.
; 5)
( )
куб.ед.
3
.
10. Solving the following improper integral
3
1
dx
I
xx
+
=
+
leads to the next result:
1) Integral diverges; 2)
1
I ln3
2
=
; 3)
1
I ln3
2
=−
;
4) I=0; 5)
1
I ln2
2
=
.
ТЕSТ № 3
1. Choose the correct answer for the integral
dx
xcos
x
+−
0
2
4
2
5
1)
1
2
5
2
+
; 2)
2
5
2
; 3)
41
2
5
2
++
;
4)
+ 4
2
5
2
π
; 5)
.
π
14
2
5
2
+−
2. Choose the correct answer for the integral
dxxe
x
−01
2
1)
2
е
−
; 2) – е; 3)
2
1 е−
; 4)
2
2 е−
; 5)
2
1 е+
.
3. Choose the correct answer for the integral
xarctgxdx
1)
( )
C
x
x
+
+
2
2
12
; 2)
C
x
arctgx
x+−+221
2
; 3)
Cx+
+21
1
;
89
4)
C
x
arctgx
x
++
+
22
1
2
; 5)
Carcctgx
x
+
2
2
.
4.
++
++
dx
ххх
хх
54
552
23
2
1) ln|x2+4x+5|–ln|x|+С; 2) ln|x2+4x+5|+arctg(x+2)–ln|x|+С;
3) 0,5ln|x2+4x+5|–arctg(x+2)+ln|x|+С; 4) 0,5ln|x2+4x+5|–arctgx+С;
5) 0,5ln|x2+4x+5|+2arctg(x+2)+ln|x|+С.
5. Choose the correct answer for the integral
dxxсos
3
4
3
1) ; 2) ; 3) ;
4) ; 5) .
6. Choose the correct answer for the integral
dx
x
x
+
9
1
8
13
1)
26−
; 2)
23−
; 3)
224 −
;
4)
24−
; 5)
21−
.
7. The area of the figure which is limited by the graphs of functions
и , is equal to:
1) 2; 2) 3; 3)
3
2
2
; 4)
3
1
3
; 5)
2
.
8. The length of the arc: y=ln(1–x2) ,from x1=0 to x2=1/2, is equal to:
1) 1/2–ln3; 2) 1–ln2; 3) ln3–1/2; 4) ln2–1; 5) 1.
9. The volume of the solid, which is generated by revolving the region
around the y-axis, where the region is bounded by y=x, y=2 and х=0, is equal to:
1)
3
2
; 2)
3
32
; 3) 4; 4)
3
8
; 5)
16
.
10. Solving the following improper integral
dx
x
x
I
−
=
2
1
1
9 3 10 2
24
−
11 10 2
24
−
33
8
9 3 11
24
−
52
4
2
48y x x= − +
2
8yx=−
90
leads to the next result:
1) Integral diverges; 2) I=2; 3) I=0,25;
4) Integral converges; 5) no solutions.
ТЕSТ № 4
1. Choose the correct answer for the integral
( )
dxхe
x
−+
2
0
32
1) е+1; 2) 2–е2; 3) е2–3; 4) 1–е; 5) 1.
2. Choose the correct answer for the integral
( )
−
−
+
1
2
3
511 x
dx
1)
72
7
; 2)
15
2
; 3)
;
72
5
4)
15
1
.
3. Choose the correct answer for the integral
+ xdxln)x( 13
2
1)
( )
Cxlnxx ++
3
; 2)
Cx ++1
2
; 3)
Cxln ++ 66
;
4)
( )
Cxxxlnxx +−−+
23
; 5)
( )
Cx
x
xlnxx +−−+
3
3
3
.
4. Choose the correct answer for the integral
dx
xx
x
++
+
102
25
2
1)
C
x
arctgхx +
+
−++
3
1
3102ln
2
5
2
;
2)
C
x
arctgхx +
+
−++
3
1
102ln5
2
;
3)
( )
Cxarctgхx ++−++ 13102ln
2
5
2
;
4)
( )
Cxarctgx ++−+ 131ln5
;
5)
Carctgxхx +−++ 3102ln
2
5
2
.
5. Choose the correct answer for the integral
1) ; 2) ; 3) ;
2 cos3сos x xdx
sin 2 sin 3
6
xx
C+
sin5 sin
10 2
хx
C++
sin sin5
25
xx
C−+
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