MATH 126
FINAL EXAM
December 14, 2005
You should do all ten problems. Read the problems carefully and answer the questions asked.
Write neatly and indicate clearly your answer to each problem. If you need more space, use
the backs of these pages and clearly indicate where the continuation may be found. You
must show your work to obtain full credit. Points may be deducted if you do not justify
your ﬁnal answer. Calculators, notes, books, or collaboration with others are
not
allowed. If
you have any questions about any of the problems ask the proctor, but no one else!
1
. [16 points] (a) Sketch the region
R
in the ﬁrst quadrant below the curve
y
= 6 +
x

x
2
.
(b) Find the volume of the solid
V
obtained by rotating the region
R
about the
y
axis.
2.
[24 points] Evaluate the integrals
(i)
Z
x
sin
xdx
(ii)
Z
3
x
2

3
x
+ 4
(
x

2)(
x
2
+ 1)
dx
(iii)
Z
x
3
√
1

x
2
dx
3
. [24 points] Determine whether or not the limit exists. If the limit exists ﬁnd it, and
indicate clearly how you obtained your answer. If the limit does not exist give reasons why.
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 Fall '07
 Mikulevicius
 Math, Power Series, Taylor Series, lim

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