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Final Exam Review Math 132 L. Ballou 1. Evaluate the following: a. 3 sin cos x dx x ´ µ ¶ . b. 2 ln x xdx ³ . c. 2 2 2 1 4 xx dx x ´ µ ¶ d. cos x xdx ³
Final Exam Review Math 132 L. Ballou 1. Evaluate the following:
a. 3sin
cosx dxx
´µ¶
.
b. 2 lnx xdx³ .
c. � �2
2
2 14
x xdx
x�
�´µ¶
d. cosx xdx³
e. � �� �2
12 4
dxx x� �
´µ¶
f. 2 16dxx �
´µ¶
g. 2
24x dxx�
´µ¶
.
h. 3 3tan secx xdx³
i. 2
lndxx x
f
µ́¶
2. Let R be the region in the first quadrant bounded by the curve 3y x and 22y x x � .
Determine the volume of the solid obtained by revolving R about a. The x-axis.
b. The y-axis.
c. The line 2x
d. The line 2y � .
3. Find the area of the region in the first quadrant bounded by 2y x , 22 4y x x � and 0y .
4. Find the length of the curve 3/2
1/2
3xy x � for 1 3xd d
5. Find the surface area when the curve 2y x for 1 2xd d is rotated about the y-axis.
6. A cylindrical tank of radius 3 ft and length of 10 ft is lying on its side on horizontal ground. If this tank is initially full of water, how much work is done to pump all the water to a point 5 ft above the top of the tank?
7. A trough has vertical ends that are equilateral triangles (downward pointing) with
sides of length 2 ft. If the trough is filled will water, find the force exerted by the water on one end of the trough.
8. Find the Taylor series for 1( )4 3
f xx
�
at 1a .
9. Find the radius and interval of convergence for the power series 21
(2 3)4
k
kk
xk
f
�¦ .
10. Determine whether the series 1 2
31
( 1)1
k
k
kk
�f
��¦ converges absolutely, converges
conditionally or diverges.
11. Determine whether the series 21
2 33 1k
kk k
f
�� �¦ converges or diverges.
12. Determine whether the series � �1
sin 51 3kk
kf
�¦ converges or diverges.
13. Find and sketch the four roots of 3 3i� . 14. Find the area of the region enclosed by one loop of the curve 3sin3r T . 15. Find the area of the region within both of the curves cosr T and 1 cosr T � .
16. Evaluate the expression
a. 213ieS
b. 5 34 2
ii
��
c. � �241 i� �
d. 3 4i� � 17. For the parametric curve 2x t , 3ln 2y t � , write the equation of the line tangent to
the curve at 1t . 18. Find the length of the parametric curve 2 1x t � , 3y t for 0 2td d .
