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20B Quizzes
Fall 2003
Bender
YOU MUST SHOW YOUR WORK.
Q1. Evaluate the following integrals.
Z
8
0
r
2
t
dt
Z
e
x
+1
dx
Q2. (a) Evaluate
Z
ln
t
t
dt
.
(b) Evaluate
R
2
1
4
x
(2
x

3)
50
dx
.
(c) Write down an integral for the area of the region enclosed by the three curves
y
=
e
x
,y
=
x
+1
,x
=2
.
Q3 (a) (8 pts) Evaluate the integrals
Z
ln
x
x
2
dx
Z
π
0
x
cos
xdx
(no trig functions in answer)
.
(b) (4 pts) Set up, but
do not evaluate
an integral for the volume obtained by
rotating the region between
y
=
x
4
and
y
= 1 about the line
y
=

2.
Q4 #1 (6 pts) Determine if the following integrals converge or diverge. Remember to
give a reason for your answer.
(a)
Z
1

1
dx
x
2
(b)
Z
∞
1
dx
x
2
.
Q4 #2 (6 pts) Estimating
R
3

1
f
(
x
)
dx
using the Trapezoidal Rule, I obtained
T
4
= 8 and
T
8
= 5. I also know that

f
0
(
x
)
≤
54 and

f
00
(
x
)
36 for

1
≤
x
≤
3.
(a) Find a guaranteed bound on the error in
T
8
.
(b) Find a reasonable estimate for the error in
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This note was uploaded on 06/17/2009 for the course MATH 20B taught by Professor Justin during the Fall '08 term at UCSD.
 Fall '08
 Justin
 Calculus, Integrals

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