Kristopher Rider Jackiewicz MAT 267 Spring 2021
Assignment Section 12.2 due 03/21/2020 at 10:00pm MST
1. (1 point) Evaluate the iterated integral I=Z1
0Z1+x
1−x
(9x2+
4y)dydx
Solution:
SOLUTION
I=Z1
0Z1+x
1−x
(9x2+4y)dydx
=Z1
09x2y+2y2y=1+x
y=1−xdx
=Z1
0h9x2(1+x) + 2(1+x)2−9x2(1−x)−2(1−x)2idx
=Z1
018x3+8xdx
=4.5x4+4x21
0
=17
2
Correct Answers:
•8.5
2. (1 point) Evaluate the iterated integral I=Z1
0Z1+y
1−y
(6y2+
6x)dxdy
Solution:
SOLUTION:
I=Z1
0Z1+y
1−y
(6y2+6x)dxdy
=Z1
06y2x+3x2x=1+y
x=1−ydy
=Z1
0h6y2(1+y) + 3(1+y)2−6y2(1−y)−3(1−y)2idy
=Z1
012y3+12ydy
=3y4+6y21
0
=9
Correct Answers:
•9
3. (1 point)
Suppose Ris the shaded region in the figure, and
f(x,y)is a continuous function on R. Find the limits
of integration for the following iterated integrals.
(a) ZZ
R
f(x,y)dA =ZB
AZD
C
f(x,y)dydx
A =
B =
C =
D =
(b) ZZ
R
f(x,y)dA =ZF
EZH
G
f(x,y)dx dy
E =
F =
G =
H =
Solution:
SOLUTION
The region Rconsists of the points on or inside the triangle
with vertices (−2,−4),(2,−4),(−2,3).
(a) The region is bounded below by the line y=−4 and above
by the line through the points (2,−4),(−2,3), which has equa-
tion y+4=−7
4(x−2).
The bounds for xare −2≤x≤2. Thus
ZZR
f(x,y)dydx =Z2
−2Z−7
4(x−2)−4
−4
f(x,y)dydx
(b) The region is bounded on the left by the line x=−2 and on
the right by the line through the points (2,−4),(−2,3), which
has equation x=−4
7(y−3)−2.
1
The bounds for yare −4≤x≤3. Thus
ZZR
f(x,y)dydx =Z3
−4Z−4
7(y−3)−2
−2
f(x,y)dydx
Correct Answers:
•-2
•2
•-4
•3-1.75*(x+2)
•-4
•3
•-2
•-[2+0.571429*(y-3)]
4. (1 point) Find the volume of the solid bounded by the
planes x=0,y=0,z=0, and x+y+z=6.
Solution:
SOLUTION
The region D, intersection of the solid with the xy-plane, is
shown below.
The region is bounded below by y=0 and above by y=6−x,
with 0 ≤x≤6. Thus the volume is given by
V=Z6
0Z6−x
0
(6−x−y)dydx
=Z6
0(6−x)y−y2
2y=6−x
y=0
dx
=Z6
0(6−x)2−(6−x)2
2dx
=Z6
0
(6−x)2
2dx
=1
2Z6
0
(6−x)2dx
Using the substitution u=6−x du =−dx, yields
V=−1
2Z0
6
u2dx
=1
2u3
36
0
=63
6
Correct Answers:
•36
5. (1 point) Consider the integral Z1
0Z8
8x
f(x,y)dydx. Sketch
the region of integration and change the order of integration.
Zb
aZg2(y)
g1(y)
f(x,y)dxdy
a=b=
g1(y) = g2(y) =
Solution:
SOLUTION
The region of integration, D, is shown below.
Because the region is
D={(x,y)|0≤x≤1,8x≤y≤8}
=(x,y)|0≤y≤8,0≤x≤y
8
we have
Z1
0Z8
8x
f(x,y)dydx =Z8
0Zy/8
0
f(x,y)dxdy
Thus a=0,b=8,g1(y) = 0 and g2(y) = y
8.
Correct Answers:
•0
•8
•0
•y/8
6. (1 point) Consider the integral Z4
0Z6√x
0
f(x,y)dydx.
Sketch the region of integration and change the order of inte-
gration.
Zb
aZg2(y)
g1(y)
f(x,y)dxdy
a=b=
g1(y) = g2(y) =
Solution:
SOLUTION
The region of integration is shown below.
Because the region is
D={(x,y)|0≤x≤4,0≤y≤6√x}
=(x,y)|0≤y≤12,y2
36 ≤x≤4
we have
Z4
0Z6√x
0
f(x,y)dydx =Z12
0Z4
y2/36
f(x,y)dxdy
Thus a=0,b=12,g1(y) = y2
36 and g2(y) = 4.
Correct Answers:
•0
•12
•yˆ2/36
2
•4
7. (1 point) Consider the integral Z6
0Z√36−y
0
f(x,y)dxdy. If
we change the order of integration we obtain the sum of two
integrals:
Zb
aZg2(x)
g1(x)
f(x,y)dydx +Zd
cZg4(x)
g3(x)
f(x,y)dydx
a=b=
g1(x) = g2(x) =
c=d=
g3(x) = g4(x) =
Solution:
SOLUTION
The region of integration is shown below.
The point Ahas coordinates (√30,6)and the curve
connecting the point (6,0)to Ahas equation y=36 −x2.
Thus
Z6
0Z√36−y
0
f(x,y)dxdy =Z√30
0Z6
0
f(x,y)dydx+Z6
√30 Z36−x2
0
f(x,y)dydx
Correct Answers:
•0
•5.47723
•0
•6
•5.47723
•6
•0
•36-xˆ2
8. (1 point) Consider the integral Z12
1Z5lnx
0
f(x,y)dydx.
Sketch the region of integration and change the order of inte-
gration.
Zb
aZg2(y)
g1(y)
f(x,y)dxdy
a=b=
g1(y) = g2(y) =
Solution:
SOLUTION
The region of integration is shown below.
The point Ahas coordinates (12,5ln(12)).
Because the region is
D={(x,y)|1≤x≤12,0≤y≤5lnx}
=n(x,y)|0≤y≤5ln(12),ey/5≤x≤12o
we have
Z12
1Z5lnx
0
f(x,y)dydx =Z5ln(12)
0Z12
ey/5f(x,y)dxdy
Thus, a=0,b=5ln(12),g1(y) = ey/5and g2(y) = 12.
Correct Answers:
•0
•12.4245
•exp(y/5)
•12
9. (1 point) In evaluating a double integral over a region D, a
sum of iterated integrals was obtained as follows:
ZZD
f(x,y)dA =Z4
0Zy
0
f(x,y)dxdy +Z8
4Z8−y
0
f(x,y)dxdy .
Sketch the region Dand express the double integral as an iter-
ated integral with reversed order of integration.
Zb
aZg2(x)
g1(x)
f(x,y)dydx
a=b=
g1(x) = g2(x) =
Solution:
SOLUTION
The region of integration is shown below.
The first integral corresponds to the region D1, while the second
integral corresponds to the region D2.
The region is bounded below by the line y=xand above by the
line y=8−x. The two lines intersect at (4,4), so 0 ≤x≤4.
Thus
Z4
0Zy
0
f(x,y)dxdy+Z8
4Z8−y
0
f(x,y)dxdy =Z4
0Z8−x
x
f(x,y)dydx
Correct Answers:
•0
•4
•x
•8-x
10. (1 point) Evaluate the integral by reversing the order of
integration.
Z1
0Z5
5y
ex2dxdy =
Solution:
SOLUTION
The region of integration is shown below.
3
The region is bounded below by y=0 and above by y=x
5, with
0≤x≤5.
Thus
Z1
0Z5
5y
ex2dxdy =R5
0R
x
5
0ex2dydx
=R5
0hyex2iy=x
5
y=0dx
=R5
0
x
5ex2dx
Using the substitution u=x2,du =2xdx, yields
=1
10 R25
0eudu
=1
10 e25 −1
Correct Answers:
•7.20049E+09
11. (1 point)
Consider the following integral. Sketch its region of
integration in the xy-plane.
Z5
0Z25
y2ysinx2dx dy
(a) Which graph shows the region of integration in
the xy-plane? [?/A/B/C/D]
(b) Write the integral with the order of integration re-
versed:
Z5
0Z25
y2ysinx2dx dy =ZB
AZD
C
ysinx2dydx
with limits of integration
A =
B =
C =
D =
(c) Evaluate the integral.
A B
C D
(Click on a graph to enlarge it)
Solution:
SOLUTION
(a) The region is bounded on the left by the function x=y2
and on the right by the vertical line x=25. The bounds for y
are 0 ≤y≤5. Thus the region corresponds to graph B.
(b) The region is bounded below by y=0 and above by y=√x,
while 0 ≤x≤25. Thus
Z5
0Z25
y2ysinx2dx dy =Z25
0Z√x
0
ysin(x2)dydx
(c)
R25
0R√x
0ysin(x2)dydx =R25
0hy2
2i√x
0sin(x2)dx
=1
2R25
0xsin(x2)dx [substitution: u=x2,du =2xdx]
=1
4R625
0sin(u)du
=1
4[−cos(u)]625
0
=1−cos(625)
4
Correct Answers:
•B
•0
•25
•0
•sqrt(x)
•[1-cos(5ˆ4)]/4
4
12. (1 point)
Consider the following integral. Sketch its region of
integration in the xy-plane.
Z0
−4Z0
−√16−x25xy dy dx
(a) Which graph shows the region of integration in
the xy-plane? [?/A/B/C/D]
(b) Evaluate the integral.
A B
C D
(Click on a graph to enlarge it)
Solution:
SOLUTION
(a) The region is bounded below by y=−√16 −x2, which
represents the lower part of a circle centered at the origin and of
radius 4, and above by y=0. Since −4≤x≤0, the region rep-
resents the quarter of a disk in the third quadrant and it matches
graph B.
(b)
Z0
−4Z0
−√16−x25xy dy dx =5Z0
−4
xy2
20
−√16−x2
dx
=−5
2Z0
−4
x(16 −x2)dx
=−5
216 x2
2−x4
40
−4
=−5
244
4−44
2
=160
Correct Answers:
•B
•160
13. (1 point) Set up a double integral in rectangular coordi-
nates for calculating the volume of the solid under the graph of
the function f(x,y) = 32 −x2−y2and above the plane z=7.
Instructions: Please enter the integrand in the first answer box.
Depending on the order of integration you choose, enter dx and
dy in either order into the second and third answer boxes with
only one dx or dy in each box. Then, enter the limits of integra-
tion.
ZB
AZD
C
A =
B =
C =
D =
Solution:
SOLUTION
The function f(x,y) = 32 −x2−y2intersects the plane z=7
when x2+y2=25.
Thus the region of integration is
D=n(x,y)| −5≤x≤5,−√25 −x2≤y≤√25 −x2o
=n(x,y)| −5≤y≤5,−p25 −y2≤x≤p25 −y2o.
The volume is then
V=R5
−5R√25−x2
−√25−x232 −x2−y2−7dydx or
V=R5
−5R√25−y2
−√25−y232 −x2−y2−7dx dy.
Correct Answers:
•25-xˆ2-yˆ2; dx; dy; -5; 5; -[sqrt(25-yˆ2)]; sqrt(25-yˆ2)
14. (1 point)
5
Suppose Ris the shaded region in the figure, and
f(x,y)is a continuous function on R. Find the limits
of integration for the following iterated integrals.
(a) ZZ
R
f(x,y)dA =ZB
AZD
C
f(x,y)dydx
A =
B =
C =
D =
(b) ZZ
R
f(x,y)dA =ZF
EZH
G
f(x,y)dx dy
E =
F =
G =
H =
Solution:
SOLUTION
The region is bounded by the circle centered at the origin with
radius 4. This circle has equation x2+y2=16. Thus the region
of integration is
R=n(x,y)| −4≤x≤4,−√16 −x2≤y≤√16 −x2o
=n(x,y)| −4≤y≤4,−p16 −y2≤x≤p16 −y2o
(a) ZZR
f(x,y)dA =Z4
−4Z√16−x2
√16−x2f(x,y)dydx
(b) ZZR
f(x,y)dA =Z4
−4Z√16−y2
√16−y2f(x,y)dx dy
Correct Answers:
•-4
•4
•-[sqrt(16-xˆ2)]
•sqrt(16-xˆ2)
•-4
•4
•-[sqrt(16-yˆ2)]
•sqrt(16-yˆ2)
15. (1 point)
Suppose Ris the shaded region in the figure, and
f(x,y)is a continuous function on R. Find the limits
of integration for the following iterated integral.
(a) ZZ
R
f(x,y)dA =ZB
AZD
C
f(x,y)dydx
A =
B =
C =
D =
Solution:
SOLUTION
The region is bounded below by the line through the points
(−3,−1),(3,0). This line has equation y=1
6(x+3)−1.
The upper bound is the line y=3, while −3≤x≤3.
Thus
ZZR
f(x,y)dA =Z3
−3Z3
1
6(x+3)−1
f(x,y)dydx
Correct Answers:
•-3
•3
•0.166667*(x+3)-1
•3
16. (1 point)
6
Consider the following integral. Sketch its region of
integration in the xy-plane.
Z2
0Ze2
ey
x
ln(x)dx dy
(a) Which graph shows the region of integration in
the xy-plane? [?/A/B/C/D]
(b) Write the integral with the order of integration re-
versed:
Z2
0Ze2
ey
x
ln(x)dx dy =ZB
AZD
C
x
ln(x)dydx
with limits of integration
A =
B =
C =
D =
(c) Evaluate the integral.
A B
C D
(Click on a graph to enlarge it)
Solution:
SOLUTION
The region is bounded on the left by the function x=eyor,
equivalently, y=lnx, and on the right by the vertical line x=e2.
The limits for yare 0 ≤y≤2. Thus
R=(x,y)|0≤y≤2,ey≤x≤e2
=(x,y)|1≤x≤e2,0≤y≤lnx
(a) The graph of this region is shown in figure A.
(b) Z2
0Ze2
ey
x
ln(x)dx dy =Ze2
1Zlnx
0
x
ln(x)dydx
(c) Ze2
1Zlnx
0
x
ln(x)dydx =Ze2
1
x dx =x2
2e2
1
=e4−1
2
Correct Answers:
•A
•1
•eˆ2
•0
•ln(x)
•[eˆ(2*2)-1]/2
17. (1 point)
Consider the following integral. Sketch its region of
integration in the xy-plane.
Z1
0Zy
√y
180x3y2dx dy
(a) Which graph shows the region of integration in
the xy-plane? [?/A/B]
(b) Evaluate the integral.
A B
(Click on a graph to enlarge it)
Solution:
SOLUTION
(a) The given integral is equivalent to −Z1
0Z√y
y
180x3y2dx dy.
Thus the region is bounded on the by left by x=yand on the
the right by x=√y, or, equivalently, y=x2,x>0.
The bounds for the variable yare 0 ≤y≤1. Thus the graph of
the region of integration matches B.
7
(b)
Z1
0Zy
√y
180x3y2dx dy =180Z1
0
y2x4
4y
√y
dy
=180
4Z1
0
y2hy4−y4/2idy
=45Z1
0y6−y4dy
=45y7
7−y5
51
0
=451
7−1
5
=−18
7
Correct Answers:
•B
•-2.57143
18. (1 point)
Find the volume of the region under the graph of f(x,y) =
2x+y+1 and above the region y2≤x, 0 ≤x≤16.
volume =
Solution:
SOLUTION
The region of integration is shown below.
Thus,
Volume =Z4
−4Z16
y2(2x+y+1)dx dy =Z4
−4
((x2)+(y+1)x)
x=16
x=y2
dy
=Z4
−4
(256 −y4)+(y+1)(16 −y2)dy.
Expanding the second binomial product, we have
Volume =Z4
−4
(256 −y4)+(16 +16y−y2−y3)dy
= (256y−y5
5) + 16(y+y2
2)−y3
3−y4
4
4
−4
= (2048 −2048
5) + 16(8)−128
3=25856
15 .
Correct Answers:
•4*2*4ˆ5/5+4*4ˆ3/3
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