Math 152 Class Notes
October 29, 2015
Review for Common Exam II
1. Evaluate the following integrals.
ˆ
(a)
dx
√
x2 9 − x2
ˆ
(b)
√
dx
x2 − 4x + 13
ˆ
(c)
x3
√
dx
9x2 − 4
ˆ
(d)
3
ˆ
(e)
4
x2
dx
x−2
x2 + 4x + 4
dx
x4 + 4x2
ˆ
2. Evaluate the integral
∞
2
1
dx or show it diverges.
x(ln x)2
ˆ
3. Using the comparison theorem, determine if the integral
1
converges or diverges.
ˆ
4. Evaluate the integral
0
3
dx
or show it diverges.
4x − 1
∞
sin2 x + e−x
dx
x3
5.Find the length of the curve parametrized by x = e3t + e−3t , y = 10 − 6t, 0 ≤ t ≤ 1.
6. A curve is given by x3 = y 2 , 1 ≤ x ≤ 4. Set up both a dx and a dy integral to nd
the following surface area.
(a) Find the surface area obtained by rotating the curve about the x-axis.
(b) Find the surface area obtained by rotating the curve about the y -axis.
2n2 + 1
7. Let an = cos
.
3n3 + 4
(a) Determine whether the sequence {an } is convergent.
(b) Determine whether the series
∞
P
an is convergent.
n=1
8. Find the sum of the following series or show it diverges
n2 + 1
.
(a) The series
an , where the sequence of partial sums is sn = 3
n +2
n=1
∞
P
(b)
∞
P
sin(nπ).
n=1
∞ 22n
P
(c)
.
n
n=1 3
9. Let an = ln(n + 1) − ln(n − 1).
(a) Determine whether the sequence {an } is convergent.
(b) Determine whether the series
∞
P
an is convergent.
n=2
10. The sequence dended by a1 =
Find the limit of the sequence.
√
2, an+1 =
√
2 + an is increasing and bounded.