CALCULUS II (Regular)
FINAL EXAM
T. FOX December 12, 2005
Write only in the answer books provided. Show all work. Do not cheat.
#1. Crunch out these integrals:
a)
3
0
1
21
x
dx
x
−+
∫ b)
22
0
cos
x
xdx
π
∫
c) 2
31
xdx
x+
∫ d) 5
sec tan
x
xdx
∫
e) 2
7
43
xdx
xx
+
++
∫ f)
1
0
1
x
dx+
∫
#2) Find the average value of
(
)
24fx x
=
+ over 05
x
≤
≤.
#3) Find the length of
()
54
4
5
f
xx= from 0
x
=
to 1
x
=
.
#4) Consider the region R bounded by 2
3yxx
=
− and 0y
=
.
a) Find the area of R.
b) Find the volume of the solid obtained by rotating R around the x-axis.
c) Find the volume of the solid obtained by rotating R around the y-axis.
#5. Determine whether these series diverge or converge. Explain carefully.
a) 3
2
1
4
6
n
n
nn
∞
=
+
+
∑ (b) 2
1
3
!
n
n
n
∞
=
+
∑ (c)
2
1
nnnn
∞
=
∑A
#6. Find the exact value of
2
21
5
n
n
n
∞
=
+
∑.
