Math 152 Class Notes
December 8, 2015
Review for Final Exam
1. Evaluate the following integrals.
ˆ
(a)
π
2
sin4 x cos5 x dx
0
ˆ
(b)
x2 arcsin x dx
ˆ
(c)
1
1
2
ˆ
(d)
dx
√
x2 9 − x2
x4 + 8
dx
x4 + 4x2
ˆ
(e)
∞
e
ˆ
(f)
0
2
dx
x(ln x)3
x
dx
1 − x2
2. Find the area bounded by x + y 2 = 2 and x + y = 0.
3. Find the average value of f (x) =
x
over the interval [1, 3].
(x + 1)2
4. A region is enclosed by the curves y = x2 and y 2 = x.
(a) Find the volume of the solid obtained by rotating the region about x-axis.
(b) Find the volume of the solid obtained by rotating the region about y -axis.
5. A curve is given by x(t) = cos3 t, y(t) = sin3 t, 0 ≤ t ≤
π
2
(a) Find the length of the curve.
(b) Find the surface area obtained by rotating the curve about the x-axis.
(c) Find the surface area obtained by rotating the curve about the y -axis.
6. Discuss the convergence of the following series:
(a) The series
∞
P
n=1
∞
P
1
(b)
cos
.
n2
n=1
(c)
∞
P
n=1
−π
5
n
∞ (ln n)2
P
(d)
.
n
n=1
.
an , where the sequence of partial sums is sn =
5 cos n
.
n
∞ (−1)n
P
convergent?
7. For what values of p is the series
p
n=2 (ln n)
8. Discuss the absolutely convergence of the following series:
(a)
∞ (−1)n
P
.
5/2 + n
n
n=1
(b)
∞ (−2)n
P
.
n=1 (2n)!
∞ (x − 1)2n
P
√
9. Find the radius and interval of convergence of the power series
.
n n+1
n=1 9
ˆ
10. Evaluate
0
1/2
cos(x2 ) − 1
dx as a innite series.
x
11. Given f (x) = ln x,
(a) Find the third degree Taylor polynomial T3 (x) for f (x) centered at a = 2.
(b) Use Taylor's Inequality to estimate the accuracy of the approximation f (x) ≈ T3 (x)
when 1 ≤ x ≤ 2.5.
12. Given the points A(−1, 3, 2), B(0, −1, 0) and C(−2, 4, 3),
(a) Find the angle 6 ABC .
(b) Find the area of the triangle 4ABC .
−→ −−→
−→
(c) Find the volume of the parallelepiped determined by OA, OB and OC .