calculus 2 assignments
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1.Evaluate each improper integral or show that it diverges.
(1)
+
1 x
dx
(2)
+
− ++ 2
2
2
xx
dx
(3)
−
2
0 2)1( x
dx
(4)
−
e
xx
dx
1 2
)(ln
1
2. For what values of k does the integral )(
)(
ab
ax
dxb
a k
−
converge and for what
values does it diverge?
3. Calculate the area of the following regions.
(1) The region bounded by xx eyey −== , and 1=x .
(2) The region bounded by )cos1(2 += a (a>0).
4. Sketch and find the area of the region bounded by cos3= and
cos1+= .
5. Find the volume of the solids generated by revolving about the x-axis the
region bounded by 02 =− yx and the parabola xy 42 = .
6. Find the length of
3
4 xy = between x = 1/3 and x = 5.
7. Find the length of the curve 41;12,23 32 −=+= ttytx .
8. Fill the blanks.
(1) If
=
−
1 2
3
n
n
n
x
a converges at 0=x , then it __________ (converges, diverges)
at 5=x .
(2) If 2lim
1
=
+
→
n
n
n c
c
, then the convergence radius of
=0
2
n
n
n xc is _______.
9. Find the convergence set of each series.
(1)
= +1
2
1
2
n
n
n
x
n
(2)
=
−+
1 3
)1(3
n
n
n
n
x
10. Find the sum function of the power series.
(1)
=1n
n
nx
(2)
=
−
−
−1
12
)11(,
12n
n
x
n
x
, and the sum of
= −1 2)12(
1
n
n
n
.
11. Find the Taylor series of
x
xf
+
=
3
1
)( in )2( −x .
12. Recall that td
t
x
x
−
=
0 2
1
1
arcsin ,
and find the first four nonzero terms in the Maclaurin series for xarcsin .
………………………………………………………………………………………
I. Fill the blanks。
1. If the series
1
1
( 1)
n
p
n n
=
− absolutely converges, then p satisfies ;
2. The convergence set of
0
1
( 3)
3
n
n
n
x
=
− is
3. The graph of +
+
=
x
dt
t
t
xf
0
21
1
)( is concave up on the interval _____________
4.
2
1
21
sin 1
1
x x
dx
x
−
+
=
+
=___________.
II. Calculations
1.
0
(1 cos 2 )
lim
tan sinx
x x
x x→
−
−
2. x
x
x
)(coslim
0+→
3. arctanx xdx
4.1
2
2
1
1x
dx
x
−
4.2
2
99
0
sin x dx
4.3
20
cos
1 sin
x
x dx
x
+
5. Find
+
−=
0
dxexI xnn , where n is a natural number.
6. Suppose that R is bounded by 3, 2, 0y x x y= = = .Find the volumes of the revolution
solids that are obtained by revolving R about x axis and y axis respectively. And the
revolution’s side area with expression.
7. Test the convergence of series
( )
( )
3
1
cos 2 1
1
2nn
n n
a
a
=
+
+
8. Find , 其中 ．
III Applications.
1. The diagram represents an equilateral triangle containing infinitely many circles,
tangent to the triangle and to neighboring circles, and reaching into the corners. What
fraction of the area of the triangle is occupied by the circles?
2. Expand into power series and find .