Math 2414 DHW 2
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University of Texas, Dallas *
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Course
2413
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
5
Uploaded by DoctorLightning3936
Ashwin Indurti
Math2414SP24
Assignment DHW
2
–
IntRev
6.2
-
S24 due 01/29/2024 at 11:59pm CST
Problem 1.
(1 point)
Consider the indefinite integral
Z
7sec
2
x
√
5
+
7tan
x
,
dx
Then the most appropriate substitution to simplify this integral is
u
=
Then
dx
=
f
(
x
)
du
where
f
(
x
)
=
After making the substitution and simplifying we obtain the inte-
gral
R
g
(
u
)
du
where
g
(
u
)
=
Evaluating the indefinite integral gives (in terms of
u
):
Replacing
u
=
f
(
x
)
in this expression gives the final answer:
Answer(s) submitted:
•
5+7tanx
•
1/(7secˆ2x)
•
1/sqrtu
•
(uˆ(1/2))/(1/2) +c
•
((5+7tanx)ˆ(1/2))/(1/2) +c
(correct)
Correct Answers:
•
5 + 7 tan(x)
•
1/(7 secˆ2(x))
•
1/sqrt(u)
•
2*sqrt(u)+C
•
2*sqrt(5+7*tan(x))+C
Problem 2.
(1 point)
Consider the definite integral
Z
6
3
2
xe
x
2
+
4
dx
Then the most appropriate substitution to simplify this integral is
u
=
Then
dx
=
f
(
x
)
du
where
f
(
x
)
=
After making the substitution and simplifying we obtain the inte-
gral
R
b
a
g
(
u
)
du
where
g
(
u
)
=
a
=
b
=
This definite integral has value =
Answer(s) submitted:
•
xˆ2+4
•
1/(2x)
•
eˆu
•
13
•
40
•
(eˆ40)-eˆ13
(correct)
Correct Answers:
•
xˆ2 + 4
•
1/(2x)
•
eˆu
•
13
•
40
•
eˆ40-eˆ13
Problem 3.
(1 point)
Evaluate the integral. (Enter ”sqrt(...)” for squar roots.)
Z
π
/
2
0
cos
(
5
x
)
dx
=
Answer(s) submitted:
•
1/5
(correct)
Correct Answers:
•
[sin(5*pi/2)]/5
1
Problem 4.
(1 point)
Evaluate the integral:
Z
5
dx
x
ln
(
2
x
)
=
Answer(s) submitted:
•
5ln|ln|2x|| + c
(correct)
Correct Answers:
•
5*ln(|ln(2*x)|)+C
Problem 5.
(1 point)
Evaluate the integral.
Z
e
x
e
x
-
1
dx
=
Answer(s) submitted:
•
ln|eˆx-1|+c
(correct)
Correct Answers:
•
ln(|exp(x)-1|)+C
Problem 6.
(1 point)
Evaluate the integral.
Z
1
/
2
0
10
x
√
1
-
x
2
dx
=
Answer(s) submitted:
•
10((-cos(arcsin(1/2)))-(-cos(0)))
(correct)
Correct Answers:
•
10*(1-[sqrt(3)]/2)
Problem 7.
(1 point)
Evaluate the integral.
Z
π
/
6
0
cos
(
x
)
sin
(
sin
(
x
))
dx
=
Answer(s) submitted:
•
-cos(1/2)-(-cos(0))
(correct)
Correct Answers:
•
1-cos(sin(pi/6))
Problem 8.
(1 point)
Evaluate the integral. (Enter ”sqrt(...)” for square roots.)
Z
-
10sin
(
√
x
)
√
x
dx
=
Answer(s) submitted:
•
20cos(sqrtx)+c
(correct)
Correct Answers:
•
20*cos(sqrt(x))+C
Problem 9.
(1 point)
Evaluate the integral.
Z
π
0
9sec
2
t
3
dt
=
Answer(s) submitted:
•
27sqrt3
(correct)
Correct Answers:
•
27*tan(pi/3)
Problem 10.
(1 point)
Evaluate the integral.
Z
10
x
1
+
x
4
dx
=
Answer(s) submitted:
•
5arctan(xˆ2)+c
(correct)
Correct Answers:
•
5*atan(xˆ2)+C
Problem 11.
(1 point)
Evaluate the integral.
Z
0
-
11
x
√
25
-
x dx
=
Answer(s) submitted:
•
-5156/15
(correct)
Correct Answers:
•
(7344-12500)/15
Problem 12.
(1 point)
Evaluate the integral.
Z
t
2
√
t
+
2
dt
=
Answer(s) submitted:
•
(2/7)(t + 2)ˆ(7/2) - (8/5)(t + 2)ˆ(5/2) + (8/3)(t + 2)ˆ(3/
(correct)
Correct Answers:
•
0.285714*(t+2)ˆ3.5-1.6*(t+2)ˆ2.5+2.66667*(t+2)ˆ1.5+C
Problem 13.
(1 point)
Find the area of the region between the curves 4
x
+
y
2
=
12 and
x
=
y
.
Area between curves =
Answer(s) submitted:
•
64/3
(correct)
Correct Answers:
•
64/3
2
Problem 14.
(1 point)
Find the area of the shaded region below.
Area =
Answer(s) submitted:
•
32-(64/3)
(correct)
Correct Answers:
•
32/3
Problem 15.
(1 point)
Find the area of the shaded region below.
Area =
Answer(s) submitted:
•
((16-4sqrt2)/3)-ln3
(correct)
Correct Answers:
•
16/3-ln(3)-4/3*sqrt(2)
Problem 16.
(1 point)
Find the volume of the solid obtained by rotating the region
bounded by
y
=
3
x
2
,
x
=
1
,
y
=
0
,
about the
x
-axis.
V
=
Answer(s) submitted:
•
pi(9/5)
(correct)
Correct Answers:
•
9*pi/5
Problem 17.
(1 point)
Find the volume of the solid obtained by rotating the region
bounded by
y
=
√
x
-
1
,
y
=
0
,
x
=
2
,
and
x
=
3 about the
x
-axis.
V
=
Answer(s) submitted:
•
pi(3/2)
(correct)
Correct Answers:
•
pi*(3ˆ2/2-3)
Problem 18.
(1 point)
Find the volume of the solid obtained by rotating the region under
the graph of the function
f
(
x
) =
2
x
-
x
2
over the interval
[
0
,
2
]
about the
x
-axis.
V
=
Answer(s) submitted:
•
pi((32/3)-(16)+(32/5))
(correct)
Correct Answers:
•
pi*2ˆ5/30
Problem 19.
(1 point)
Find the volume of the solid obtained by rotating the region under
the graph of the function
f
(
x
) =
2
x
+
1
over the interval
[
0
,
2
]
about
the
x
-axis.
V
=
Answer(s) submitted:
•
(8pi)/3
(correct)
Correct Answers:
•
-4*pi/(2+1)+4*pi
Problem 20.
(1 point)
Find the volume of the solid obtained by rotating the region under
the graph of the function
f
(
x
) =
e
x
over the interval
[
0
,
5
]
about
the
x
-axis.
V
=
Answer(s) submitted:
•
((pieˆ(10))/2)-pi/2
(correct)
Correct Answers:
•
pi/2*[eˆ(2*5)-1]
3
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Problem 21.
(1 point)
Find the volume of the solid obtained by rotating the region
bounded by
x
=
6
y
2
,
y
=
1
,
and
x
=
0
,
about the
y
-axis.
V
=
Answer(s) submitted:
•
pi(36/5)
(correct)
Correct Answers:
•
6*6*pi/5
Problem 22.
(1 point)
Find the volume of the solid obtained by rotating the region
bounded by
y
=
2
x
+
6
,
y
=
0, and
x
=
0 about the
y
-axis. (Hint:
The region is a triangle to the left of the
y
-axis.)
V
=
Answer(s) submitted:
•
18pi
(correct)
Correct Answers:
•
pi/3*2*3ˆ3
Problem 23.
(1 point)
Find the volume of the solid obtained by rotating the region
bounded by the curves
y
=
sec
(
x
)
,
y
=
1
,
x
=
-
1
,
and
x
=
1 about
the
x
-axis.
V
=
Answer(s) submitted:
•
pi(2tan(1)-2)
(correct)
Correct Answers:
•
pi*2*[tan(1)-1]
Problem 24.
(1 point)
Find the volume of the solid formed by rotating the region en-
closed by
y
=
x
2
,
y
=
4
x
in QI about the
x
-axis.
V
=
Answer(s) submitted:
•
(2048pi)/15
(correct)
Correct Answers:
•
pi*[1/3*4ˆ2*4ˆ3-5ˆ(-1)*4ˆ5]
Problem 25.
(1 point)
Find the volume of the solid obtained by rotating the region en-
closed by
y
=
√
100
-
x
2
and
y
=
8 about the
x
-axis.
V
=
Answer(s) submitted:
•
288pi
(correct)
Correct Answers:
•
4*216/3*pi
Problem 26.
(1 point)
Find the volume of the solid obtained by rotating the region
bounded by the curves
y
=
x
2
,
y
=
0
,
x
=
0, and
x
=
2
,
about the
y
-axis.
V
=
Answer(s) submitted:
•
25.133
(correct)
Correct Answers:
•
pi*4ˆ2/2
Problem 27.
(1 point)
Using disks or washers, find the volume of the solid obtained by
rotating the region enclosed by the curves
x
=
y
2
and
x
=
3
y
about
the
y
-axis.
V
=
Answer(s) submitted:
•
pi((81-(243/5)))
(correct)
Correct Answers:
•
pi*2/15*3ˆ5
Problem 28.
(1 point)
Find the volume of the solid formed by rotating the region en-
closed by
x
=
8
y
and
x
=
y
3
(with
y
≥
0 ) about the
y
-axis.
V
=
Answer(s) submitted:
•
((64/3)*(16*sqrt(2)) - ((2*sqrt(2))ˆ7)/7)pi
(correct)
Correct Answers:
•
4*pi/21*8ˆ(7/2)
4
Problem 29.
(1 point)
Find the volume of the solid obtained by rotating the region en-
closed by the graphs
x
=
y
2
and
x
=
2
√
y
in the first quadrant about
the
y
-axis.
V
=
Answer(s) submitted:
•
(3/10)pi
(correct)
Correct Answers:
•
pi*[2/(2+2)-1/(2*2+1)]
Generated by c WeBWorK, http://webwork.maa.org, Mathematical Association of America
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