Mei Qin Chen
citadel-math231
WeBWorK assignment number Homework15 is due : 10/25/2011 at 09:00am EDT.
The
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for the course contains the syllabus, grading policy and other information.
This file is /conf/snippets/setHeader.pg you can use it as a model for creating files which introduce each problem set.
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having trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA’s or your professor for
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Give 4 or 5 significant digits for (floating point) numerical answers. For most problems when entering numerical answers,
you can if you wish enter elementary expressions such as 2 ∧ 3 instead of 8, sin(3 ∗ pi/2)instead of -1, e ∧ (ln(2)) instead of 2,
(2 + tan(3)) ∗ (4 − sin(5)) ∧ 6 − 7/8 instead of 27620.3413, etc. Here’s the list of the functions which WeBWorK understands.
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1. (1 pt) Library/272/setStewart15 2/problem 2.pg
Evaluate the iterated integral
R3R2
0
0
6. (1 pt) Library/272/setStewart15 2/problem 6.pg
Calculate the volume under the elliptic paraboloid z = 2x2 + 3y2
and over the rectangle R = [−1, 1] × [−4, 4].
6x2 y3 dxdy
Correct Answers:
Correct Answers:
• 324
• 266.666666666667
2. (1 pt) Library/272/setStewart15
2/problem 3.pg
R R
Evaluate the iterated integral 23 23 (4x + y)−2 dydx
Correct Answers:
• 0.00657932707934328
7. (1 pt) Library/272/setStewart15 3/problem 3.pg
Find the volume of the solid bounded by the planes x = 0, y = 0,
z = 0, and x + y + z = 5.
3. (1 pt) Library/272/setStewart15 1/ur vc 8 8.pg
Using the maxima and minima of the function, produce upper
and Zlower
Z estimates of the integral
Correct Answers:
5 cos(x − y)dA where R is the region: R = [0, 1] × [0, 1].
I=
R
≤I≤
• 20.8333333333333
Correct Answers:
• 2.7015115293407
• 5
4. (1 pt) Library/272/setStewart15 2/problem 12.pg
Evaluate the integral
R π/6 R 4
0
2
8. (1 pt) Library/FortLewis/Calc3/16-2-Iterated-integrals/HGM4-162-14-Iterated-integrals.pg
(y cos x − 3) dydx.
Correct Answers:
Consider the following integral. Sketch its region of
integration in the xy-plane.
• -0.141592653589794
Z 1Z y
5. (1 pt) Library/272/setStewart15 R3/problem
1.pg
R 1+x
Evaluate the iterated integral I = 01 1−x
(21x2 + 2y) dydx
0
√ 150x
y
2 3
y dx dy
(a) Which graph shows the region of integration in
the xy-plane? ?
Correct Answers:
• 12.5
(b) Evaluate the integral.
1
A
B
A
B
C
D
(Click on a graph to enlarge it)
Correct Answers:
• B
• -1.94805
(Click on a graph to enlarge it)
Correct Answers:
• D
• 10
9. (1 pt) Library/FortLewis/Calc3/16-2-Iterated-integrals/HGM4-162-23-Iterated-integrals.pg
10. (1 pt) Library/FortLewis/Calc3/16-2-Iterated-integrals/HGM4-162-19-Iterated-integrals.pg
Consider the following integral. Sketch its region of
integration in the xy-plane.
Consider the following integral. Sketch its region of
integration in the xy-plane.
Z 0Z 0
√
−2 −
4−x2
Z 1.5Z x
5xy dy dx
0
(a) Which graph shows the region of integration in
the xy-plane? ?
2
ex dy dx
0
(a) Which graph shows the region of integration in
the xy-plane? ?
(b) Evaluate the integral.
(b) Evaluate the integral.
2
Consider the following integral. Sketch its region of
integration in the xy-plane.
Z 2Z 4
y sin x2 dx dy
0
y2
(a) Which graph shows the region of integration in
the xy-plane? ?
(b) Write the integral with the order of integration reversed:
Z BZ D
Z 2Z 4
y sin x2 dy dx
y sin x2 dx dy =
A
0
B
y2
A
C
with limits of integration
(Click on a graph to enlarge it)
A=
B=
C=
D=
(c) Evaluate the integral.
Correct Answers:
• A
• 4.24387
A
B
C
D
(Click on a graph to enlarge it)
Correct Answers:
•
•
•
•
•
•
11. (2 pts) Library/FortLewis/Calc3/16-2-Iterated-integrals/HGM416-2-30-Iterated-integrals.pg
3
B
0
4
0
sqrt(x)
(1 - cos(2ˆ4))/4
• ln(x)
• ( e**(2*1) - 1)/2
12. (2 pts) Library/FortLewis/Calc3/16-2-Iterated-integrals/HGM416-2-38-Iterated-integrals.pg
13. (1 pt) Library/Michigan/Chap16Sec2/Q13.pg
Consider the shaded region in the graph below.
Consider the following integral. Sketch its region of
integration in the xy-plane.
Z 1 Z e1
x
dx dy
y
ln(x)
0 e
(a) Which graph shows the region of integration in
the xy-plane? ?
R
Write R f dA on this region as an iterated integral:
RbRd
f
d ,
R dA = a c f (x, y) d
where
a=
,
b=
,
, and
c=
d=
.
SOLUTION
This region lies between x = 0 and x = 6 and between the
lines y = 18 − 3x and y = 18, and so the iterated integral is
(b) Write the integral with the order of integration reversed:
Z 1 Z e1
Z BZ D
x
x
dx dy =
dy dx
y
ln(x)
ln(x)
0 e
A C
with limits of integration
R
A=
B=
C=
D=
Z 6 Z 18
f (x, y) dy dx.
(c) Evaluate the integral.
0
18−3x
Alternatively, we could have set up the integral as follows:
Z 18 Z 6
0
18−y
3
f (x, y) dx dy.
Correct Answers:
A
•
•
•
•
•
•
B
y
x
0
6
18 - 3*x
18
14. (1 pt) Library/Michigan/Chap16Sec2/Q17.pg
Consider the shaded region in the graph below.
R
C
Write R f dA on this region as an iterated integral:
RbRd
f
d
where
R dA = a c f (x, y) d
a=
,
b=
,
c=
, and
d=
.
SOLUTION
The line connecting (1, 0) and (3, 2) is
R
D
(Click on a graph to enlarge it)
Correct Answers:
•
•
•
•
C
1
e
0
y = x − 1.
4
So the integral is
•
•
•
•
•
•
Z 3Z 4
f (x, y) dy dx.
1
x−1
Note that for this region it is not possible to write
single integral with dA = dx dy.
R
R
f dA as a
Correct Answers:
c
Generated by the WeBWorK system WeBWorK
Team, Department of Mathematics, University of Rochester
5
y
x
1
3
2*(x-1)/2
4