Math 2414 DHW 3
.pdf
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School
University of Texas, Dallas *
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Course
2413
Subject
Mathematics
Date
Apr 3, 2024
Type
Pages
3
Uploaded by DoctorLightning3936
Ashwin Indurti
Math2414SP24
Assignment DHW
3
–
6.3
-
7.1
–
S24 due 02/05/2024 at 11:59pm CST
Problem 1.
(1 point)
Find the volume of the solid obtained by rotating the region
bounded by
y
=
1
x
,
x
=
2
,
x
=
8
,
and
y
=
0 about the
y
-axis.
V
=
Answer(s) submitted:
•
37.699
(correct)
Problem 2.
(1 point)
Use the method of cylindrical shells to find the volume of the solid
obtained by rotating the region bounded by the curves
y
=
8
x
2
,
y
=
0
,
x
=
1
,
and
x
=
2 about the
y
-axis.
V
=
Answer(s) submitted:
•
60pi
(correct)
Problem 3.
(1 point)
Use the method of cylindrical shells to find the volume of the solid
generated by rotating the region bounded by
y
=
e
-
9
x
2
,
y
=
0
,
x
=
0
,
and
x
=
8 about the
y
-axis.
V
=
Answer(s) submitted:
•
(pi(eˆ576-1))/(9eˆ576)
(correct)
Problem 4.
(1 point)
Find the volume of the solid obtained by rotating the region
bounded by
y
=
4cos
(
x
2
)
,
y
=
0,
x
=
0, and
x
=
r
π
3
about the
y
-axis.
V
=
Answer(s) submitted:
•
(4pisqrt3)/2
(correct)
Problem 5.
(1 point)
Find the volume of the solid obtained by rotating the region
bounded by the curves
x
=
√
y
,
x
=
0
,
and
y
=
9 about the
x
-axis.
V
=
Answer(s) submitted:
•
(972pi)/5
(correct)
Problem 6.
(1 point)
Find the volume of the solid obtained by rotating the region
bounded by the curves
x
=
1
+
y
2
,
x
=
0
,
y
=
3
,
and
y
=
4 about
the
x
-axis.
V
=
Answer(s) submitted:
•
(189pi)/2
(correct)
Problem 7.
(1 point)
Use the method of cylindrical shells to find the volume of the solid
obtained by rotating the region bounded by the curves
x
=
4
y
2
-
y
3
and
x
=
0 about the
x
-axis.
V
=
Answer(s) submitted:
•
(512pi)/5
(correct)
Problem 8.
(1 point)
Use the shell method to compute the volume of the solid obtained
by rotating the region enclosed by the functions
y
=
x
2
,
y
=
2
-
x
2
and to the right of
x
=
0 about the
y
-axis.
V
=
Answer(s) submitted:
•
pi
(correct)
Problem 9.
(1 point)
Use the method of cylindrical shells to find the volume generated
by rotating the region bounded by
y
=
4
(
x
-
2
)
2
and
y
=
x
2
-
4
x
+
7
about the
y
-axis.
V
=
Answer(s) submitted:
•
16pi
(correct)
Problem 10.
(1 point)
Let
V
be the volume of the solid obtained by rotating about the
y
-axis the region bounded by
y
=
2
√
x
and
y
=
2
x
2
.
Find
V
either
by the disk/washer method or by cylindrical shells.
V
=
Answer(s) submitted:
•
(3pi)/5
(correct)
1
Problem 11.
(1 point)
Evaluate the integral.
Z
x
cos
(
2
x
)
dx
=
Answer(s) submitted:
•
(1/4)(2xsin(2x) + cos(2x) + c)
(correct)
Problem 12.
(1 point)
Evaluate the integral.
Z
xe
4
x
dx
=
.
Answer(s) submitted:
•
1/16(4x(eˆ(4x))-eˆ(4x))+c
(correct)
Problem 13.
(1 point)
Evaluate the integral.
Z
4
1
-
3
√
t
ln
(
t
)
dt
=
Answer(s) submitted:
•
28/3 - 32ln2
(correct)
Problem 14.
(1 point)
Evaluate the integral:
Z
1
0
-
4
y
e
2
y
dy
=
Answer(s) submitted:
•
(3/eˆ2)-1
(correct)
Problem 15.
(1 point)
Suppose that
f
(
1
) =
10
,
f
(
4
) =
-
4
,
f
0
(
1
) =
-
9
,
f
0
(
4
) =
-
8
,
and
f
00
is continuous. Use this to evaluate the integral.
Z
4
1
x f
00
(
x
)
dx
=
Answer(s) submitted:
•
-32+23
(correct)
Problem 16.
(1 point)
Evaluate the integral:
Z
5
t
3
e
t
dt
=
Answer(s) submitted:
•
5((tˆ3*eˆt)-3((tˆ2*eˆt)-2((eˆt*t)-eˆt)))+c
(correct)
Problem 17.
(1 point)
Evaluate the integral.
Z
1
0
5
(
x
2
+
1
)
e
-
x
dx
=
Answer(s) submitted:
•
5((-6/e)+3)
(correct)
Problem 18.
(1 point)
Evaluate the integral.
Z
π
/
2
π
/
4
3
x
csc
2
(
x
)
dx
=
Answer(s) submitted:
•
-3((-pi/4)-((1/2)ln2))
(correct)
Problem 19.
(1 point)
Evaluate the integral.
Z
-
2
x
2
sin
(
π
x
)
dx
=
Answer(s) submitted:
•
-2((-1/pi)xˆ2cos(pix)+ (2/pi)((1/pi)xsin(pix)+(1/piˆ2)cos(
(correct)
Problem 20.
(1 point)
Evaluate the integral.
Z
x
sec
2
(
7
x
)
dx
=
Answer(s) submitted:
•
1/2xˆ2secˆ2(7x)-1/98(49xˆ2tanˆ2(7x)-14xtan(7x)-2ln|cos(7x)
(correct)
Problem 21.
(1 point)
Evaluate the integral.
Z
sin
-
1
(
5
x
)
dx
=
Answer(s) submitted:
•
xarcsin(5x)+1/5sqrt(1-25xˆ2)+c
(correct)
2
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