Quiz 2 - Math XXXXXXXXXX)
Instructions:
• The quiz is worth 100 points. There are 10 problems, each worth 10 points. Your score
on the quiz will be converted to a percentage and posted in your assignment folder with
comments.
• This quiz is open book and open notes, and you may take as long as you like on it
provided that you submit the quiz no later than the due date posted in our course
schedule of the syllabus. You may refer to your textbook, notes, and online classroom
materials, but you may not consult anyone.
• You must show all of your work to receive full credit. If a problem does not seem to
equire work, write a sentence or two to justify your answer.
• Please write neatly. Illegible answers will be assumed to be inco
ect.
• Please remember to show ALL of your work on every problem. Read the basic rules fo
showing work below BEFORE you start working on the quiz:
1. Each step should show the complete expression or equation rather than a piece of
it.
2. Each new step should follow logically from the previous step, following rules of
alge
a.
3. Each new step should be beneath the previous step.
4. The equal sign, =, should only connect equal numbers or expressions.
• This quiz is due at 11:59 PM (Eastern Time) on Tuesday, November 8.
***********************
At the end of your quiz you must include the following dated statement with
your name typed in lieu of a signature. Without this signed statement you will
eceive a zero.
I have completed this quiz myself, working independently and not consulting anyone except
the instructor. I have neither given nor received help on this quiz.
Name: Date:
1
Quiz 2 - Volume, Arc Length, Surface Area
1. Find the volume of the solid obtained by rotating about the x-axis the region under the
curve y =
√
x from 0 to 1. Sketch the solid and explain how you form the integral that
calculates the volume of the solid.
2. Find the volume of a pyramid whose base is a square with side ` and whose height is
h. Sketch the solid and indicate how you form the integral.
3. Find the volume of the solid obtained by rotating the region bounded by y = x3, y = x,
and x ≥ 0 about the x-axis. Sketch the bounded region which will be rotated about
the x-axis.
4. Find the exact length of the curve y =
1
4
x2 − 1
2
lnx, where 1 ≤ x ≤ 2.
5. Find the exact length of the curve y =
x3
3
+
1
4x
, where 1 ≤ x ≤ 2. (See chapter 5-2 fo
eference)
6. Find the exact length of the curve y = ln(sec x), where 0 ≤ x ≤ π
4
. (See chapter 5-2
for reference)
7. Find the area of the surface obtained by rotating the curve y =
√
1 + 4x, where 1 ≤
x ≤ 5, about the x-axis. (Use chapter 5-2)
8. Chapter 5-3, Problem 32.
9. Chapter 5-3, Problem 34.
10. Chapter 5-4, Problem 14.
2
definite integral. Partition the time interval [a,b] into short subintervals. For the interval [ty 4]:
force = f(c;) for any c; in [ti_y. tj]
distance moved =
work = fej) \| (Axi/Ay
Toulwok = 3 work slong cach subimenat} = 3 16) var (ay? &
=H
J 0 \ an? + @y/a0® dt = total work along the path (x(0. 300)
ta
In problems XXXXXXXXXX, find the total work along the given parametric path. If necessary. approximate the value
of the integral using your calculator. f is in pounds, x and y are in feet, i tes.
316) = 1. x0) = cos. y(0 =sin().0< 12%. 32. {0 =1. xO =ty0 = 0=r=1.
Ban =r. x0 =r yn =1.0<1<1 34. £0) = sin(n. x(0)=20,y(1) =31,0 r= x. (Fig.27)
35. 1()=1. x() = cos(n) y(t) = sin), 0s 12x (Fig XXXXXXXXXXCan you find a geometric way to calculate the
shaded area?)
mporary Calculus
In problems 11-26, sketch the region bounded between the given functions on the interval and calculate
the centroid of each region (use Simpson's rule with n = 20 if necessary). Plot the location of the centroid
on your skeich of the region
.y=x and the x-axis for 0.
. y=x? and the x-axis for 2x2.
y=x> and the line y=4 for 2
y=sin(x) and the x-axis for 0.x <7.
y=4-x> and the x-axis for 25x52 x2 and y=x for 0sx<1
y=9-x and y=3 for 0x3, y=
y=VR andthe x-axis for 0x9. 20. y= In(x) and the x-axis for 1
y=¢* andthe line for 0=x<1 22. y=x% and the 2x for 0x22
An empty one foot square tin box (Fig. 25) weighs 10 pounds and its center of
mass is 6 inches above the bottom of the box. When the box is full with 60
pounds of liquid. the center of mass of the box-liquid system is again 6 inches
of mass of the box-liquid
d in the box.
(b) What height of liquid in the bottom of the box results in the box-liqui
system having the lowest center of mass (and the greatest stability)?