Math 2414 DHW 7
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University of Texas, Dallas *
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Course
2413
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
4
Uploaded by DoctorLightning3936
Ashwin Indurti
Math2414SP24
Assignment DHW
7
–
8.1
8.2
9.3
–
S24 due 03/04/2024 at 11:59pm CST
Problem 1.
(1 point)
Consider the curve defined by the equation
y
=
5
x
4
+
17
x
.
Set
up an integral that represents the length of curve from the point
(
-
3
,
354
)
to the point
(
3
,
456
)
.
Z
dx
Answer(s) submitted:
•
3
•
-3
•
sqrt(1+(20xˆ3+17)ˆ2)
(correct)
Correct Answers:
•
3
•
-3
•
sqrt(1+(20*xˆ3+17)ˆ2)
Problem 2.
(1 point)
Use the arc length formula to find the length of the curve
y
=
5
-
3
x
,
-
3
≤
x
≤
2.
You can check your answer by noting the shape of the curve.
Arc length =
Answer(s) submitted:
•
5sqrt10
(correct)
Correct Answers:
•
5*sqrt(10)
Problem 3.
(1 point)
Use the arc length formula to find the length of the curve
y
=
√
4
-
x
2
, 0
≤
x
≤
2
.
You can check your answer by noting the
shape of the curve.
Arc length =
Answer(s) submitted:
•
pi
(correct)
Correct Answers:
•
2*pi/2
Problem 4.
(1 point)
Find the exact length of the curve
x
=
2
3
(
y
2
-
1
)
3
/
2
, 1
≤
y
≤
4
.
Arc length =
Answer(s) submitted:
•
39
(correct)
Correct Answers:
•
1/3*(1-3*4+2*4ˆ3)
Problem 5.
(1 point)
Find the exact length of the curve
y
=
ln
(
cos
x
)
from
x
=
0 to
x
=
π
/
6
.
Length:
Answer(s) submitted:
•
1/2ln3
(correct)
Correct Answers:
•
ln(sec(pi/6)+tan(pi/6))
Problem 6.
(1 point)
Find the exact length of the curve
x
2
=
11
y
3
between the points
(
0
,
0
)
and
(
121
,
11
)
.
Length:
Answer(s) submitted:
•
(incorrect)
Correct Answers:
•
8/(27*11)*[(9/4*11ˆ2+1)ˆ(3/2)-1]
Problem 7.
(1 point)
Find the exact length of
y
=
1
4
x
2
-
1
2
ln
x
over the interval
[
1
,
8
e
]
.
Length:
Answer(s) submitted:
•
1/4(64eˆ2+2ln(8e)-1)
(correct)
Correct Answers:
•
(8ˆ2*eˆ2+1)/4+[ln(8)]/2
Problem 8.
(1 point)
Find the exact length of the curve
y
=
x
3
6
+
1
2
x
,
1
2
≤
x
≤
1.
Arc length =
Answer(s) submitted:
•
31/48
(correct)
Correct Answers:
•
31/48
1
Problem 9.
(1 point)
Find the arc length of the curve
y
=
1
2
(
e
x
+
e
-
x
)
from
x
=
0 to
x
=
3
.
Length:
Answer(s) submitted:
•
(eˆ6-1)/(2eˆ3)
(correct)
Correct Answers:
•
1/2*[eˆ3-eˆ(-3)]
Problem 10.
(1 point)
Which of the following integrals represents the area of the surface
obtained by rotating the curve
y
=
e
x
,
1
≤
y
≤
2
,
about the
y
-axis?
•
A. 2
π
Z
2
1
y
q
1
+(
1
/
y
)
2
dy
•
B. 2
π
Z
2
1
e
y
q
1
+(
1
/
y
)
2
dy
•
C. 2
π
Z
2
1
ln
(
y
)
p
1
+(
1
/
y
)
dy
•
D. 2
π
Z
2
1
e
y
p
1
+(
1
/
y
)
dy
•
E. 2
π
Z
2
1
ln
(
y
)
q
1
+(
1
/
y
)
2
dy
•
F. 2
π
Z
2
1
y
p
1
+(
1
/
y
)
dy
Answer(s) submitted:
•
B
(incorrect)
Correct Answers:
•
E
Problem 11.
(1 point)
Which of the following integrals represents the area of the surface
obtained by rotating the curve
y
=
ln
(
x
)
,
1
≤
x
≤
3
,
about the
x
-axis?
•
A. 2
π
Z
3
1
x
q
1
+(
1
/
x
)
2
dx
•
B. 2
π
Z
3
1
ln
(
x
)
p
1
+(
1
/
x
)
dx
•
C. 2
π
Z
3
1
ln
(
x
)
q
1
-
(
1
/
x
)
2
dx
•
D. 2
π
Z
3
1
ln
(
x
)
q
1
+(
1
/
x
)
2
dx
•
E. 2
π
Z
3
1
x
q
1
+(
1
/
x
)
2
dx
•
F. 2
π
Z
3
1
x
q
1
+(
1
/
x
)
2
dx
Answer(s) submitted:
•
D
(correct)
Correct Answers:
•
D
Problem 12.
(1 point)
Find the area of the surface generated by rotating
y
=
4
x
+
3,
2
≤
x
≤
4, about the
x
-axis.
S
=
Answer(s) submitted:
•
pi(60sqrt17)
(correct)
Correct Answers:
•
pi*sqrt(17)*60
Problem 13.
(1 point)
Find the area of the surface generated by rotating the curve
x
=
4
+
4
y
2
,
, 0
≤
y
≤
1 about the
x
-axis.
S
=
Answer(s) submitted:
•
-pi/96+(65sqrt(65)pi)/96
(correct)
Correct Answers:
•
2*pi/(3*64)*[65ˆ(3/2)-1]
2
Problem 14.
(1 point)
Find the area of the surface generated by rotating the curve
y
=
2
x
3
from
x
=
0 to
x
=
1 about the
x
-axis.
S
=
Answer(s) submitted:
•
(pi(37sqrt37-1))/54
(correct)
Correct Answers:
•
pi/(27*2)*[(1+9*2ˆ2)ˆ(3/2)-1]
Problem 15.
(1 point)
Find the area of the surface generated by rotating the portion of
the curve 9
x
=
y
2
+
18
,
2
≤
x
≤
6
,
in Quadrant I about the
x
-axis.
S
=
Answer(s) submitted:
•
49pi
(correct)
Correct Answers:
•
49*pi
Problem 16.
(1 point)
Find the area of the surface obtained by rotating the curve
y
=
√
6
x
from
x
=
0 to
x
=
7 about the
x
-axis.
S
=
Answer(s) submitted:
•
-6pi+(34sqrt51*pi)/3
(correct)
Correct Answers:
•
pi*sqrt(6)/6*[(4*7+6)ˆ(3/2)-6ˆ(3/2)]
Problem 17.
(1 point)
Find the area of the surface generated by rotating the curve
x
=
2
√
5
-
y
,
-
1
≤
y
≤
0 about the
y
-axis.
S
=
Answer(s) submitted:
•
-16sqrt6*pi+(56sqrt7*pi)/3
(correct)
Correct Answers:
•
8*pi/3*[7ˆ(3/2)-6ˆ(3/2)]
Problem 18.
(1 point)
Find the area of the surface generated by revolving the curve
x
=
p
9
-
y
2
,
-
1
≤
y
≤
1 about the
y
-axis.
S
=
Answer(s) submitted:
•
12pi
(correct)
Correct Answers:
•
12*pi
Problem 19.
(1 point)
Find the area of the surface obtained by rotating the curve
x
=
1
3
(
y
2
+
2
)
3
/
2
,
0
≤
y
≤
1
,
about the
x
-axis.
S
=
Answer(s) submitted:
•
(3pi)/2
(correct)
Correct Answers:
•
2*pi*(1ˆ2/2+1ˆ4/4)
Problem 20.
(1 point)
Find the area of the surface obtained by rotating the curve
y
=
x
3
6
+
1
2
x
,
1
2
≤
x
≤
1
,
about the
x
-axis.
S
=
Answer(s) submitted:
•
263pi/256
(correct)
Correct Answers:
•
263/256*pi
Problem 21.
(1 point)
Find the general explicit solution of the differential equation
y
0
=
3
xe
x
2
.
y
=
Answer(s) submitted:
•
3/2eˆ(xx)+c
(correct)
Correct Answers:
•
1.5*eˆ(xˆ2)+C
Problem 22.
(1 point)
Find the general implicit solution of the differential equation
(
8
+
x
4
)
dy
dx
=
x
3
y
y
2
=
Answer(s) submitted:
•
1/2ln|8+xˆ4|+c
(correct)
Correct Answers:
•
0.5*ln(|8+xˆ4|)+C
3
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Problem 23.
(1 point)
Find the explicit solution of the initial value problem
dy
dx
+
y
cos
(
x
) =
4cos
(
x
)
,
y
(
0
) =
6
y
=
Answer(s) submitted:
•
2eˆ(-sinx)+4
(correct)
Correct Answers:
•
4 + 2*eˆ(-sin(x))
Problem 24.
(1 point)
Solve the initial value problem
dy
dx
=
2
x
+
1
18
y
,
y
(
0
) =
-
3
y
=
.
Answer(s) submitted:
•
-sqrt(xˆ2+x)/3
(incorrect)
Correct Answers:
•
-(1/3) sqrt(xˆ2 + x + 81)
Problem 25.
(1 point)
Find the general explicit solution of the differential equation
dP
dt
=
4
P
+
a
.
where
a
is a non-zero constant.
P
=
Answer(s) submitted:
•
4t+a
(incorrect)
Correct Answers:
•
-a/4+A*eˆ(4*t)
Problem 26.
(1 point)
Find the general explicit solution of the differential equation
dR
dx
=
a
(
R
2
+
16
)
.
where
a
is a non-zero constant.
R
=
Answer(s) submitted:
•
4tan(4ax)
(correct)
Correct Answers:
•
4*tan(4*(a*x+C))
Problem 27.
(1 point)
Solve the initial value problem. Assume
-
π
2
<
y
<
π
2
xy
0
+
7
y
0
=
3cos
2
(
y
)
,
y
(
-
6
) =
π
4
y
=
.
Answer(s) submitted:
•
(incorrect)
Correct Answers:
•
arctan(3 ln|x+7|+1)
Problem 28.
(1 point)
Solve the initial value problem
dy
dx
=
2
xy
+
1
x
-
4
y
-
2
,
y
(
0
) =
4
2
y
=
.
Answer(s) submitted:
•
(incorrect)
Correct Answers:
•
(1/2)(5*eˆ(1 xˆ2 - 4 x)-1)
Generated by c WeBWorK, http://webwork.maa.org, Mathematical Association of America
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