Section 7.2: Volumes
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified
line. Sketch the region, the solid, and a typical disk or washer.
2.
y = 1 x 2 ,
4.
y =ln x ,
y =0 ; about the x-axis.
y =1 ,
Anthony Kitsmiller
Assignment Section 9.1 due 04/26/2015 at 11:58pm MST
Zhu MAT 266 ONLINE B Spring 2015
3. (1 pt)
Consider the parametric curve:
1. (1 pt)
Eliminate the parameter t to nd a Cartesian equation for:
2
2
The curve is (part of ) an ellipse an
C. HECKMAN Solutions, 12:15 Class
MAT 266 Test 1
Solutions to the 1:40 classs version of the test appear ve pages later.
(1) [15 points] Do a trigonometric substitution on the following integral. That is, make a substitution which gets rid of the
MAT 270
Learning to Find Antiderivatives Exercises
Each of the following is an accumulation function. For each one, find the closed-
form rate of change with respect to x. (This is just like the mastery exam; you
MAT 266 Homework
Name_
A meteor approaching the atmosphere was estimated to have a speed function of
s(t) = 432 7.5t , where s is the speed in m / sec and time t is the time in seconds since
the meteor was at a specified point in space.
The goal is to use
1
MAT 266 Test 1 Review
Techniques of integration and improper integrals, sections 5.5, 6.1-6.4 and 6.6 in
Essential Calculus, Early Transcendentals, 2nd Edition, by James Stewart.
Find the following indenite integrals (1-21).
1.
4.
(ln x)5
dx
5x
2.
1
dx
MAT 266
Test2
MAT 266
TEST 2 - ANSWERS
SoMSS, ASU
Directions:
1. There are 14 questions worth a total of 60 points.
2. Questions 1 - 10 are Multiple Choice worth 4 points each to be answered on
the supplied SCANTRONS.
3. Questions 11 - 14 are Free Respons
MAT 266
Test3
MAT 266
TEST 3 ANSWERS/SOLUTIONS
SoMSS, ASU
Directions:
1. There are 14 questions worth a total of 60 points.
2. Questions 1 - 10 are Multiple Choice worth 4 points each to be answered on
the supplied SCANTRONS.
3. Questions 11 - 14 are Free
MAT 266
Test2
MAT 266
TEST 2
SoMSS, ASU
Directions:
1. There are 14 questions worth a total of 60 points.
2. Questions 1 - 10 are Multiple Choice worth 4 points each to be answered on
the supplied SCANTRONS.
3. Questions 11 - 14 are Free Responses worth 5
1.
2.
3.
A 10-ft chain weighs 25 lb and hangs from a ceiling. Find the work done (in ft-lb) in lifting the
lower end of the chain to the ceiling so that it is level with the upper end.
The center of mass (CM) of the hanging chain is at 5 ft. The CM of the
FINAL EXAM PRACTICE
Trig Review
1. Complete the table:
0
6
3
4
2
sin()
cos()
2. Find all values of , 0 < 2 so that
1
1
(a) cos() = 2
(b) sin() = 2
(c) cos(3) = 0
I. Tangent lines to parametric curves.
1. Find an equation of the tangent line to the curve a
Practice for Test 3, MAT 266
Determine whether the following series converges absolutely. Justify your answer with the proper series test.
2n !
n!
( 1)n (n 5)!
( 2)n 3 n 3
1.
2.
3.
4. n 2
n
n
n
5! n!4
3
n 0 3 n!
n 0
n 1
n 0 1,000,000
Determine the radi
1
MAT 266 Test 2 Review
Approximate Integration, Applications of Integrals, Sequences and Series; sections 6.5,
7.1-7.4, 7.6, 8.1, 8.2 in Essential Calculus, Early Transcendentals, 2nd Edition, by James Stewart.
21
1) Estimate 1
using (write all answers
1
MAT 266 Test 1 Review
Techniques of integration and improper integrals, sections 5.5, 6.1-6.4 and 6.6 in
Essential Calculus, Early Transcendentals, 2nd Edition, by James Stewart.
Find the following indenite integrals (1-21).
1.
4.
(ln x)5
dx
5x
2.
1
dx
Taylor and Maclaurin Series
In the preceding section you were able to find power series representations for a certain restricted
class of functions. Here we will investigate which functions have power series representations
and how we can find such repres
Parametric Curves
A curve C consisting of points (x,y) where x and y are both functions of a third
variable, say t (called a parameter). Thus, we have each point (x,y) determined by
the parametric equations x f= g ( t ) .
=
( t ) and y
Note: Often, but no
Sequences (addition to other notes)
A. Since sequences are functions with domains restricted to the natural numbers the function
limit theorems hold for convergent sequence. Therefore we use the function techniques
with sequences.
Do the following sequenc
1
Inverted circular cone pool
A tank has the shape of an inverted circular cone with height 10 m and base
radius 4 m. It is lled with water to a height of 8 m. Find the work required
to empty the tank by pumping all of the water to the top of the tank. Th
Power Series
Definition:
If c0 , c1 , c2 , c3 ,. are constants and x is a variable, then a series of the form
c x
n =0
n
n
= c1 x + c2 x 2 + c3 x3 + c4 x 4 + . + cn x n + . is called a power series in x.
c0 +
For each fixed value of x, the series is a ser
Chapter 2 {Sequences}
Solve for
(a) .x2x=2 (b) sinx=x
Sketch and state the domain.
(a) f (x) = x2 (b) g(vx) = _C_.
x+l
Sketch the above functions if the domains are restricted to the Natural Numbers. Notation: _
@ %=# w) ¢=n
arm: or {H
or {n2} or
Deniti
Tuesday May 18
Name_
Suppose this shows the rate of change of an insect population in insects/month over the
course of 1.8 months.
1) Explain what's happening to the insect population.
MAT 266 Homework
Name
A meteor approaching the atmosphere was estimated to have a speed function of
s(t) = 432 — 7.51, where s is the speed in m 1’ sec and time t is the time in seconds since
the meteor was at a specified point in space.
The goal is to
Exact Rate of Change Functions: Rules for Closed Forms
Inverse Trig
Basic
Note: sin 1 x = arcsin x
Trig
f (x)
f (x)
d
f (x)
dx
c
nx n1
d
f (x)
dx
sin 1 x
sin x
cos x
cos 1 x
cos x
sin x
tan 1 x
1
1+ x 2
cot 1 x
0
xn
d
f (x)
dx
x
1
2 x
ax
a x ln a
ex
ex
1
Homework for Tues May 22
Name_
Suppose this shows the rate of change of an insect population in insects/month over the course of
1.8 months.
1) Explain what's happening to the insect population.
a) .in the first 0.2 months
b) .from 0.2 to 0.6 months
(No c
Thomas Andrews
MAT 266
Homework 1
Answers:
5.5
1
(16 t)4 dt
#4
U=1-6t Du=-6dt
1 1
du
6 u4
1
1 3
u
6
3
1 3
u +c
18
1
(16 t)3+ c
18
#14
x
( x 2+1 )
2
dx
2
u=x +1 du=2 x dx
1 1
du
2 u2
1
1 u1 +c
2
1 2 1
( x +1 ) +c
2
sin x x dx
#18
Thomas Andrews
Mat 266
Thomas Andrews
MAT 266
Assignment zero
Questions
1: The graded activities homework and quizzes, tests, and the final exam.
2: The two term tests will be on Test 1 (Wen 9/21), Test 2 (Wen 10/19), and Test 3
(Wen 11/16). The final exam will be on Tuesday, D