MAT 202 Midterm, Fall 2009
American University of Science and Technology · Calculus II · 2 pages · Ms. Hiba Othman
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MAT 102/202
1/2
1) (10 pts)Find the length of the curve:
t
x
9cos3
and
t
y
9sin3
when
4
0
t
2) (10 pts) Find the cosine of the angle between the 2 vectors:
j
i
u
4
3
and
2
2
j
i
v
3) (15 pts – 5 pts each)Solve the following
a)
dx
x
cos 3ln x
b)
x
e
dx
2
c)
dx
x
x
3
1
1
4) (10 pts)Evaluate
using logarithmic differentiation
4
5
3
5
2
1
2
2
.
1
2
3
tan
cos
3
e x
x
x
x
x
x
x
y
5) (15 pts – 5 pts each)Find
of the following functions
a)
xy
y
ex
1
cos
2
b)
4
ln
5
ln
3
2
x
x e
y
x
c)
2
2
3
3 2
5
y
xy
6) (10 pts)Find the volume generated by revolving the region bounded by
the curves
e
y
ex
y
,
and x=0 about the x-axis.
7) (10 pts)Find the length of the curve:
8
ln
2
x
x
y
when
2
1
x
MAT 102/202
2/2
8) (20 pts – 10 pts each)
a) Find the volume of the solid generated by rotating the region
3
0
y x
and
1
0
x
about the line
y 1
b) Compute the volume of the solid generated by revolving the triangular
region bounded by the lines
4
2
y x
,
y x
and
x 0
about the line
x 4
using the Shell Method
BONUS:(10 PTS)
Solve the following integrals
a)
4
1
2
2
ln
tan
e
dx
x
x
b)
dx
x
x
2
2
tan
1
sec