Calculus 2
1. Evaluate each improper integral or show that it diverges.
(1) +
1 x
dx
(2) +
− ++ 22 2
xx
dx
(3) −
2
0 2)1( x
dx
(4) −
e
xx
dx
1 2 )(ln1
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 .
