MATH 102 Final, Fall 2005
American University of Beirut · Calculus and Analytical Geometry II · 2 pages
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Math 102
Final Exam
Dr. H. Yamani
Fall 2005
Name:
Do each of the following problems. Show all your work. No work shown. No credit.
****GOOD LUCK****
1. Evaluate the following integrals:
(5 points each)
a.
dx
e
x
1
1
+
∫
b
∫
dx
x
x
17
8
3
2
+
+
c.
dx
x
x
x
x
∫
+
+
−
2
2
3
1
d.
dx
x
x
∫
+
+
1
9
2
2
e.
∫
−
+
−
3
4
2
x
x
dx
2. Evaluate the following integrals using trigonometric substitution: (5 points each)
a.
∫
(
)
dx
x
3 / 2
2
1
1
+
b.
∫
−
2
16
x
dx
3. Evaluate the following improper integrals:
(5 points each)
a.
ln x dx
∫
1
0
b.
dx
x
x
x
∫
∞
−
−
2
2
3
1
3
4. Test the following integrals for convergence or divergence. Explain. (5 points each)
a.
(
)
dx
x
∫
∞
+
1
9
9
7
b.
(
)
dx
x
x
1 / 3
1
3
1
+
∫
∞
5. Find the area of the region bounded between the curve
x
x
(f x )
+
=
2
and the line
g( x ) = x + 4
. (5 points)
6. Use the washer method to find the volume of the solid generated by revolving the
region bounded by
2x
y =
and the line
y = 1
about the line
y = 2
.
( 5 points)
7. Use Logarithmic differentiation to find the derivative of y with respect to x .(5 points)
(
)
4
3
3
1
1
/
x
x
x
y
−
+
=
8. Find an equation for the plane through the point (1,2,3) parallel to
u = 2i + 3j + k and v = i j + 2k
(5 points)
9. Find the area of a triangle with vertices P(2 ,-1, 4) , Q( 1, 0, -1) , R(1, 2, 3) (5 points)
10. Find parametric equations for the line through P(1, 2, 5) and Q(-1, 3, 2). (5 points)
11. Find the angle between the planes
0
2
3
=
+
−
z
y
x
and
= 1
+
+
z
y
x
(5 points)
12. Determine the following limit:
(5 points)
x
x
x
lim
−
→∞
2
1
1
13. Find the center and radius of the sphere.
(5 points)
9
2
2
3
3
3
2
2
2
=
−
+
+
+
z
y
z
y
x