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Unformatted text preview: Math 21B (Winter 2006)
Kouba
Exam 2 KEY Please PRINT your name here : _________________________________________________________ __
Your Exam ID Number __________ __ 1. PLEASE DO NOT TURN THIS PAGE UNTIL TOLD TO DO SO. 2. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY,
ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. IT IS A VIO
LATION OF THE UNIVERSITY HONOR CODE TO COPY ANSWERS FROM SOME
ONE ELSE’S EXAM. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH
AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED
OR DISTRACTED. THANK YOU FOR YOUR COOPERATION. 3. No notes, books, or classmates may be used as resources for this exam. YOU MAY
USE A CALCULATOR ON THIS EXAM. 4. Read directions to each problem carefully. Show all work for full credit. In most
cases, a correct answer with no supporting work will NOT receive full credit. What you
write down and how you write it are the most important means of your getting a good
score on this exam. Neatness and organization are also important. 5. Make sure that you have 6 pages, including the cover page.
6. You will be graded on proper use of integral and derivative notation.
7. Include units on answers where units are appropriate. 8. You have until 11:50 am. to ﬁnish the exam. 1.) (12 pts.) The base of a solid lies in the region bounded by the graphs of y =2 272, y = 0,
and a: = 2. Crosssections of the solid taken perpendicular to the :caxis at :r are rectangles
of height 5. SET UP BUT DO NOT EVALUATE an integral which represents the volume of this solid.
M M y yzxk
Agx): (M)(Ju47jvt)
: VM 2 S:AL><) M 1
S :80 53(LM 101 ’* (AW/Le 2.) (12 pts.) A ﬂat plate of variable density lies in the region bounded by the graphs of
y = 3+sinx, y = 0, a: = O, and a: = 27r. Density at point (x, y) is given by 6(50, y) = 3+\/E.
SET UP BUT DO NOT EVALUATE integrals which represent 37, the ycoordinate for the
center of mass of this plate. 2 § 1% (3+Mx)%C3+VY) A1
,_ W
“V “ 3:“ C3+MX}(3+13Z]M 3.) (6 pts. each) Consider the region bounded by the graphs of y = In x, y = 0, and x = e.
SET UP BUT DO NOT EVALUATE integrals which represent the volume of the solid formed by revolving this region about a.) the xaxis using the DISC METHOD. 8, Vol 2 Tr 8‘ (foM b.) the yaxis using the SHELL METHOD. M: if STCX)@~LX) M
. J (W e.) the line a: = —1 using the DISC METHOD. VO/Q: TrSLCeA—Ol ~ T 3' (6)14”! '2
f 0: °(* )Jg d.) the line y = 2 using the SHELL METHOD. Vvl = NT SLCWJ) (64y) d1 2‘ . 4.) (13 pts.) Set up and EVALUATE an integral which represents the arc length of the
following curve, which is given parametrically by : {x=(1/3)t3 ,
y=(1/2)t2,forOSt§1. Am: 31, (emery ow 5.) (13 pts.) A chain weighs 2 pounds per foot and is used to raise a 100 pound object
on the ground to a. point 50 feet above the ground. Set up and EVALUATE an integral
which represents the work required to complete the task (Include the weight of the chain in your solution.) ‘ y:50©# u
\3
O
G
o
l
X)
ox
o
O
n
V
0‘
O
0
I 6.) (13 pts.) A ﬂat plate is in the shape of a right triangle with legs 3 feet and 4 feet. It
rests vertically on its 4—foot edge at the bottom of a pool ﬁlled to a depth of 8 feet. SET UP BUT DO NOT EVALUATE an integral which represents the force (on one side) of
water pressure on this plate. (Water weighs 62.4 pounds per cubic foot.) 2% CMXWMW) : (C%y)cly)C8—>’) (6.2.4) /. P: $0 (excl). (w g—Yjﬁyyj a;Z
L64. 7.) (13 pts.) The semicircle y = \/1 — $2 for —1 S a: S 1 is rotated about the xaxis to
form a sphere. Set up and EVALUATE an integral which represents the surface area of
this sphere. The following EXTRA CREDIT problem is OPTIONAL. It is worth 10 points. 1.) A circle has radius a units. Its center is b units from line L. The circle is rotated about
line L to form a torus (doughnut). Use integration to compute the volume of the torus. L y xz+yz:6\9~ ...
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This note was uploaded on 09/30/2008 for the course MATH 21B taught by Professor Vershynin during the Winter '08 term at UC Davis.
 Winter '08
 Vershynin
 Calculus

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