تكلم مع الذكاء الاصطناعي
Deck 17: Line and Surface Integrals
ملء الشاشة (f)
سؤال
Define 
where 
Compute the vector assigned to the point 
by the vector field 
where Compute the vector assigned to the point by the vector field 
سؤال
Let 
be the curve 
.
A) Parametrize
using polar coordinates.
B) Find the length of
be the curve . A) Parametrize
using polar coordinates. B) Find the length of
سؤال
Define 
At what points is 
normal to the vector 
At what points is normal to the vector سؤال
Let 
. Express 
in terms of the unit radial vector 
in 
.
. Express in terms of the unit radial vector in . سؤال
Compute the vector assigned to the point 
by the vector field 
by the vector field سؤال
Let 
be a differentiable function of r, and let 
.
Express
in terms of the unit radial vector 
.
be a differentiable function of r, and let .Express
in terms of the unit radial vector . سؤال
Find a potential function for the field 
by inspection.
by inspection. سؤال
Calculate the work performed by the force field 
on a particle that moves in the counterclockwise direction around the quarter circle 
. Assume that 
is in Newtons and the unit of distance is the meter.
on a particle that moves in the counterclockwise direction around the quarter circle . Assume that is in Newtons and the unit of distance is the meter. سؤال
Find a potential function for 
by inspection.
by inspection. سؤال
Compute the work performed in moving a particle along the path 
by the force 
.
by the force . سؤال
Define 
and 
Compute 
and Compute سؤال
Define 
Compute the vector assigned to the point 
by the vector field 
Compute the vector assigned to the point by the vector field سؤال
Compute 
, where 
is the part of the ellipse 
joining the point 
to the point 
and 
.
, where is the part of the ellipse joining the point to the point and . سؤال
Let 
. Express 
in terms of the unit radial vector 
in 
.
. Express in terms of the unit radial vector in . سؤال
Let 
. Express 
in terms of the unit radial vector 
in 
.
. Express in terms of the unit radial vector in . سؤال
Let 
be the triangle with vertices at the points 
and 
in counterclockwise order.
Compute the line integral
.
be the triangle with vertices at the points and in counterclockwise order.Compute the line integral
. سؤال
Determine whether the vector field 
is a conservative vector field, and if so, find a potential function for 
by inspection.
is a conservative vector field, and if so, find a potential function for by inspection. سؤال
Compute 
, where 
is the part of the parabola 
starting at 
and ending at 
.
, where is the part of the parabola starting at and ending at . سؤال
Let 
denote the closed curve of intersection of the hemisphere 
and the cylinder 
oriented counterclockwise.
Compute
where 
.
denote the closed curve of intersection of the hemisphere and the cylinder oriented counterclockwise.Compute
where . سؤال
Evaluate 
, where 
is the rectangle in 
with vertices at 
, and 
, in this order.
, where is the rectangle in with vertices at , and , in this order. سؤال
Let 
.
A) Determine whether
is conservative, and if so, find a potential function for 
B) Compute the integral
where C is the path 
from 
to 
.
. A) Determine whether
is conservative, and if so, find a potential function for B) Compute the integral
where C is the path from to . سؤال
Let 
. Find a function 
so that 
is conservative in 
and 
for all 
.
. Find a function so that is conservative in and for all . سؤال
Evaluate 
for the path C shown in the following figure. The path consists of the line segment from (1, 0, 0) to (0, 1, 0), followed by the circular arc in the yz-plane with radius 1 joining (0, 1, 0) to (0, 0, 1). 
for the path C shown in the following figure. The path consists of the line segment from (1, 0, 0) to (0, 1, 0), followed by the circular arc in the yz-plane with radius 1 joining (0, 1, 0) to (0, 0, 1). سؤال
Find 
where 
is the path 
from 
to 
and 
.
where is the path from to and . سؤال
Compute the work performed in moving a particle along the path 
by the force 
by the force سؤال
Compute the work performed in moving a particle along the path 
by the force 
by the force سؤال
Compute the length of the parametric curve 
, 
, سؤال
Compute the line integral 
where C is the segment from 
to 
and 
.
where C is the segment from to and . سؤال
Let 
.
Determine whether
is conservative. If so, find a potential function.
.Determine whether
is conservative. If so, find a potential function. سؤال
Compute 
where C is the quarter circle on the 
plane starting at 
and ending at 
where C is the quarter circle on the plane starting at and ending at سؤال
Evaluate 
where C is the parametric curve 
from 
to 
.
where C is the parametric curve from to . سؤال
Compute the mass of the curve 
if the mass density is 
and 
are measured in centimeters.
if the mass density is and are measured in centimeters. سؤال
Let 
. Compute 
where 
is the segment joining the points 
and 
together with the quarter circle centered at the origin and joining the points 
and 
. The integration is performed from 
to 
.
. Compute where is the segment joining the points and together with the quarter circle centered at the origin and joining the points and . The integration is performed from to . سؤال
Let 
A) Compute
B) Evaluate
where C is the parametric curve 
from 
to 
A) Compute
B) Evaluate
where C is the parametric curve from to سؤال
Let C be the curve 
, 
, 
, 
, and let 
.
The value of
is which of the following?
A)
B)
C)
D)
E) None of the answers is correct.
, , , , and let .The value of
is which of the following?A)
B)
C)
D)
E) None of the answers is correct.
سؤال
Compute the line integral 
where C is the line segment from 
to 
and 
.
where C is the line segment from to and . سؤال
Compute the mass of the curve 
if the mass density is 
if the mass density is سؤال
Compute 
where 
and 
is the path 
from 
to 
.
where and is the path from to . سؤال
Evaluate 
where C is the line segment joining the points 
and 
where C is the line segment joining the points and سؤال
Compute the work performed in moving a particle along the path 
by the force 
by the force سؤال
Find 
where 
is the path 
from 
to 
and 
where is the path from to and سؤال
Let 
.
Compute
where C is the spiral 
.Compute
where C is the spiral سؤال
Let S denote the part of the surface 
inside the cylinder 
The area of S is closest to which of the following?
A)
B)
C)
D)
E)
inside the cylinder The area of S is closest to which of the following?A)
B)
C)
D)
E)
سؤال
In the paraboloid 
, 
the charge density is equal to the distance from the 
plane. Find the total charge in the paraboloid.
, the charge density is equal to the distance from the plane. Find the total charge in the paraboloid. سؤال
Let 
.
A) Determine whether
is conservative, and if so, find a potential function for 
B) Compute the line integral
where C is the path consisting of: 
: the parabola 
from 
to 
: the line segment from 
to 
and 
: the semicircle in the plane 
with diameter 
where 
and 
. A) Determine whether
is conservative, and if so, find a potential function for B) Compute the line integral
where C is the path consisting of: : the parabola from to : the line segment from to and : the semicircle in the plane with diameter where and سؤال
Compute 
along the curve 
shown in the following figure. 
along the curve shown in the following figure. سؤال
Consider the vector field 
Find the formula for 
so that 
is conservative and 
Find the formula for so that is conservative and سؤال
Consider the vector field 
Find the formula for 
so that 
is conservative and 
Find the formula for so that is conservative and سؤال
Let 
.
A) Does
have a potential function? If so, find it.
B) Find the integral
where C is the semicircle oriented counterclockwise with the diameter 
, where 
and 
. 
. A) Does
have a potential function? If so, find it. B) Find the integral
where C is the semicircle oriented counterclockwise with the diameter , where and . سؤال
Let 
.
A) Determine whether
has a potential function, and if so, find a potential function for 
.
B) Compute
where 
, 
is the curve 
, 
from 
to 
and 
is the straight line from 
to 
. A) Determine whether
has a potential function, and if so, find a potential function for . B) Compute
where , is the curve , from to and is the straight line from to سؤال
Compute 
where S is the part of the plane 
which lies inside the half cylinder 
.
where S is the part of the plane which lies inside the half cylinder . سؤال
Find 
where 
is the path 
from 
to 
and 
.
where is the path from to and . سؤال
Find 
where 
is the path 
from 
to 
and 
where is the path from to and سؤال
Consider the vector field 
Find the formula for 
so that 
is conservative and 
Find the formula for so that is conservative and سؤال
Compute the surface integral 
for the surface 
.
for the surface . سؤال
The surface integral 
where 
is the part of the plane 
which lies inside the cylinder 
is which of the following?
A)
B)
C)
D) 0
E) None of the answers is correct.
where is the part of the plane which lies inside the cylinder is which of the following?A)
B)
C)
D) 0
E) None of the answers is correct.
سؤال
The mass of the part of the cone 
with 
and density 
is which of the following?
A)
B)
C)
D)
E) None of the answers is correct.
with and density is which of the following?A)
B)
C)
D)
E) None of the answers is correct.
سؤال
Find the surface area of the surface 
.
. سؤال
Evaluate the surface integral 
where S is the part of the cone 
between the planes 
and 
.
where S is the part of the cone between the planes and . سؤال
Find 
where 
is the path 
from 
to 
and 
where is the path from to and سؤال
Let S be the surface 
.
The area of S is approximately which of the following
A)
B)
C)
D)
E) None of the answers is correct.
.The area of S is approximately which of the following
A)
B)
C)
D)
E) None of the answers is correct.
سؤال
The integral 
where 
and 
with normals pointing to the positive y direction is equal to which of the following?
A)
B)
C)
D)
E)
where and with normals pointing to the positive y direction is equal to which of the following?A)
B)
C)
D)
E)
سؤال
Compute 
where S is the part of the plane 
in the first octant bounded by the planes 
and 
where S is the part of the plane in the first octant bounded by the planes and سؤال
Evaluate 
where 
is the portion of the paraboloid 
contained within the cylinder 
where is the portion of the paraboloid contained within the cylinder سؤال
Evaluate 
where S is the surface cut from the paraboloid 
by the planes 
, 
and 
. 
where S is the surface cut from the paraboloid by the planes , and . سؤال
Evaluate 
where 
and 
is the surface 
oriented outward.
where and is the surface oriented outward. سؤال
Evaluate 
where 
is the portion of the cylinder 
bounded between the planes 
and 
where is the portion of the cylinder bounded between the planes and سؤال
Find the flux of 
across the surface 
, oriented outward.
across the surface , oriented outward. سؤال
Compute 
where S is the part of the surface 
between the planes 
and 
where S is the part of the surface between the planes and سؤال
Compute the flux of water through the parabolic cylinder (oriented in the positive y-direction)
S :
, 
, 
where the velocity vector is 
(in 
).
S :
, , where the velocity vector is (in ). سؤال
Compute the area of the surface 
.
. سؤال
Find the surface area of the portion of the sphere 
satisfying 
satisfying سؤال
Let 
where 
and 
. Let 
be a sector of angle 
in the sphere of radius 
, centered at the origin and oriented outward. Find 
. 
where and . Let be a sector of angle in the sphere of radius , centered at the origin and oriented outward. Find . سؤال
The area of the surface 
is which of the following?
A)
B)
C)
D)
E) None of the answers is correct.
is which of the following?A)
B)
C)
D)
E) None of the answers is correct.
سؤال
Find the flux of the field 
through the surface 
which is the part of the paraboloid 
where 
.
through the surface which is the part of the paraboloid where . سؤال
Compute the surface integral 
where 
and 
is the surface of the sphere centered at the origin with radius 4.
where and is the surface of the sphere centered at the origin with radius 4. سؤال
Find the surface area of the portion of the cone 
within the cylinder 
within the cylinder سؤال
Find the flux 
of the field 
through the surface 
oriented upward.
of the field through the surface oriented upward. سؤال
Parametrize the surface 
first using Cartesian coordinates and then using cylindrical coordinates. Assume x and y take on all real values.
first using Cartesian coordinates and then using cylindrical coordinates. Assume x and y take on all real values. سؤال
Find the area of the ellipse cut from the plane 
by the cylinder 
.
by the cylinder .
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العب
ملء الشاشة (f)
Deck 17: Line and Surface Integrals
1
Define where Compute the vector assigned to the point by the vector field
where Compute the vector assigned to the point by the vector field 
2
Let be the curve .
A) Parametrize using polar coordinates.
B) Find the length of
be the curve . A) Parametrize
using polar coordinates. B) Find the length of
A) 
B) 8
B) 8 3
Define At what points is normal to the vector
At what points is normal to the vector 
4
Let . Express in terms of the unit radial vector in .
. Express in terms of the unit radial vector in . فتح الحزمة
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5
Compute the vector assigned to the point by the vector field
by the vector field فتح الحزمة
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6
Let be a differentiable function of r, and let .
Express in terms of the unit radial vector .
be a differentiable function of r, and let .Express
in terms of the unit radial vector . فتح الحزمة
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7
Find a potential function for the field by inspection.
by inspection. فتح الحزمة
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8
Calculate the work performed by the force field on a particle that moves in the counterclockwise direction around the quarter circle . Assume that is in Newtons and the unit of distance is the meter.
on a particle that moves in the counterclockwise direction around the quarter circle . Assume that is in Newtons and the unit of distance is the meter. فتح الحزمة
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9
Find a potential function for by inspection.
by inspection. فتح الحزمة
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10
Compute the work performed in moving a particle along the path by the force .
by the force . فتح الحزمة
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11
Define and Compute
and Compute فتح الحزمة
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12
Define Compute the vector assigned to the point by the vector field
Compute the vector assigned to the point by the vector field فتح الحزمة
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13
Compute , where is the part of the ellipse joining the point to the point and .
, where is the part of the ellipse joining the point to the point and . فتح الحزمة
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14
Let . Express in terms of the unit radial vector in .
. Express in terms of the unit radial vector in . فتح الحزمة
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15
Let . Express in terms of the unit radial vector in .
. Express in terms of the unit radial vector in . فتح الحزمة
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16
Let be the triangle with vertices at the points and in counterclockwise order.
Compute the line integral .
be the triangle with vertices at the points and in counterclockwise order.Compute the line integral
. فتح الحزمة
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17
Determine whether the vector field is a conservative vector field, and if so, find a potential function for by inspection.
is a conservative vector field, and if so, find a potential function for by inspection. فتح الحزمة
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18
Compute , where is the part of the parabola starting at and ending at .
, where is the part of the parabola starting at and ending at . فتح الحزمة
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19
Let denote the closed curve of intersection of the hemisphere and the cylinder oriented counterclockwise.
Compute where .
denote the closed curve of intersection of the hemisphere and the cylinder oriented counterclockwise.Compute
where . فتح الحزمة
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20
Evaluate , where is the rectangle in with vertices at , and , in this order.
, where is the rectangle in with vertices at , and , in this order. فتح الحزمة
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21
Let .
A) Determine whether is conservative, and if so, find a potential function for
B) Compute the integral where C is the path from to .
. A) Determine whether
is conservative, and if so, find a potential function for B) Compute the integral
where C is the path from to . فتح الحزمة
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22
Let . Find a function so that is conservative in and for all .
. Find a function so that is conservative in and for all . فتح الحزمة
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23
Evaluate for the path C shown in the following figure. The path consists of the line segment from (1, 0, 0) to (0, 1, 0), followed by the circular arc in the yz-plane with radius 1 joining (0, 1, 0) to (0, 0, 1).
for the path C shown in the following figure. The path consists of the line segment from (1, 0, 0) to (0, 1, 0), followed by the circular arc in the yz-plane with radius 1 joining (0, 1, 0) to (0, 0, 1). فتح الحزمة
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24
Find where is the path from to and .
where is the path from to and . فتح الحزمة
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25
Compute the work performed in moving a particle along the path by the force
by the force فتح الحزمة
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26
Compute the work performed in moving a particle along the path by the force
by the force فتح الحزمة
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27
Compute the length of the parametric curve ,
, فتح الحزمة
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28
Compute the line integral where C is the segment from to and .
where C is the segment from to and . فتح الحزمة
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29
Let .
Determine whether is conservative. If so, find a potential function.
.Determine whether
is conservative. If so, find a potential function. فتح الحزمة
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30
Compute where C is the quarter circle on the plane starting at and ending at
where C is the quarter circle on the plane starting at and ending at فتح الحزمة
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31
Evaluate where C is the parametric curve from to .
where C is the parametric curve from to . فتح الحزمة
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32
Compute the mass of the curve if the mass density is and are measured in centimeters.
if the mass density is and are measured in centimeters. فتح الحزمة
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33
Let . Compute where is the segment joining the points and together with the quarter circle centered at the origin and joining the points and . The integration is performed from to .
. Compute where is the segment joining the points and together with the quarter circle centered at the origin and joining the points and . The integration is performed from to . فتح الحزمة
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34
Let
A) Compute
B) Evaluate where C is the parametric curve from to
A) Compute
B) Evaluate
where C is the parametric curve from to فتح الحزمة
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35
Let C be the curve , , , , and let .
The value of is which of the following?
A)
B)
C)
D)
E) None of the answers is correct.
, , , , and let .The value of
is which of the following?A)
B)
C)
D)
E) None of the answers is correct.
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36
Compute the line integral where C is the line segment from to and .
where C is the line segment from to and . فتح الحزمة
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37
Compute the mass of the curve if the mass density is
if the mass density is فتح الحزمة
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38
Compute where and is the path from to .
where and is the path from to . فتح الحزمة
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39
Evaluate where C is the line segment joining the points and
where C is the line segment joining the points and فتح الحزمة
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40
Compute the work performed in moving a particle along the path by the force
by the force فتح الحزمة
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41
Find where is the path from to and
where is the path from to and فتح الحزمة
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42
Let .
Compute where C is the spiral
.Compute
where C is the spiral فتح الحزمة
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43
Let S denote the part of the surface inside the cylinder The area of S is closest to which of the following?
A)
B)
C)
D)
E)
inside the cylinder The area of S is closest to which of the following?A)
B)
C)
D)
E)
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44
In the paraboloid , the charge density is equal to the distance from the plane. Find the total charge in the paraboloid.
, the charge density is equal to the distance from the plane. Find the total charge in the paraboloid. فتح الحزمة
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45
Let .
A) Determine whether is conservative, and if so, find a potential function for
B) Compute the line integral where C is the path consisting of: : the parabola from to : the line segment from to and : the semicircle in the plane with diameter where and
. A) Determine whether
is conservative, and if so, find a potential function for B) Compute the line integral
where C is the path consisting of: : the parabola from to : the line segment from to and : the semicircle in the plane with diameter where and فتح الحزمة
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46
Compute along the curve shown in the following figure.
along the curve shown in the following figure. فتح الحزمة
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47
Consider the vector field Find the formula for so that is conservative and
Find the formula for so that is conservative and فتح الحزمة
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48
Consider the vector field Find the formula for so that is conservative and
Find the formula for so that is conservative and فتح الحزمة
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49
Let .
A) Does have a potential function? If so, find it.
B) Find the integral where C is the semicircle oriented counterclockwise with the diameter , where and .
. A) Does
have a potential function? If so, find it. B) Find the integral
where C is the semicircle oriented counterclockwise with the diameter , where and . فتح الحزمة
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50
Let .
A) Determine whether has a potential function, and if so, find a potential function for .
B) Compute where , is the curve , from to and is the straight line from to
. A) Determine whether
has a potential function, and if so, find a potential function for . B) Compute
where , is the curve , from to and is the straight line from to فتح الحزمة
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51
Compute where S is the part of the plane which lies inside the half cylinder .
where S is the part of the plane which lies inside the half cylinder . فتح الحزمة
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52
Find where is the path from to and .
where is the path from to and . فتح الحزمة
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53
Find where is the path from to and
where is the path from to and فتح الحزمة
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54
Consider the vector field Find the formula for so that is conservative and
Find the formula for so that is conservative and فتح الحزمة
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55
Compute the surface integral for the surface .
for the surface . فتح الحزمة
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56
The surface integral where is the part of the plane which lies inside the cylinder is which of the following?
A)
B)
C)
D) 0
E) None of the answers is correct.
where is the part of the plane which lies inside the cylinder is which of the following?A)
B)
C)
D) 0
E) None of the answers is correct.
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57
The mass of the part of the cone with and density is which of the following?
A)
B)
C)
D)
E) None of the answers is correct.
with and density is which of the following?A)
B)
C)
D)
E) None of the answers is correct.
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58
Find the surface area of the surface .
. فتح الحزمة
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59
Evaluate the surface integral where S is the part of the cone between the planes and .
where S is the part of the cone between the planes and . فتح الحزمة
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60
Find where is the path from to and
where is the path from to and فتح الحزمة
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61
Let S be the surface .
The area of S is approximately which of the following
A)
B)
C)
D)
E) None of the answers is correct.
.The area of S is approximately which of the following
A)
B)
C)
D)
E) None of the answers is correct.
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62
The integral where and with normals pointing to the positive y direction is equal to which of the following?
A)
B)
C)
D)
E)
where and with normals pointing to the positive y direction is equal to which of the following?A)
B)
C)
D)
E)
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63
Compute where S is the part of the plane in the first octant bounded by the planes and
where S is the part of the plane in the first octant bounded by the planes and فتح الحزمة
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64
Evaluate where is the portion of the paraboloid contained within the cylinder
where is the portion of the paraboloid contained within the cylinder فتح الحزمة
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65
Evaluate where S is the surface cut from the paraboloid by the planes , and .
where S is the surface cut from the paraboloid by the planes , and . فتح الحزمة
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66
Evaluate where and is the surface oriented outward.
where and is the surface oriented outward. فتح الحزمة
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67
Evaluate where is the portion of the cylinder bounded between the planes and
where is the portion of the cylinder bounded between the planes and فتح الحزمة
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68
Find the flux of across the surface , oriented outward.
across the surface , oriented outward. فتح الحزمة
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69
Compute where S is the part of the surface between the planes and
where S is the part of the surface between the planes and فتح الحزمة
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70
Compute the flux of water through the parabolic cylinder (oriented in the positive y-direction)
S : , , where the velocity vector is (in ).
S :
, , where the velocity vector is (in ). فتح الحزمة
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71
Compute the area of the surface .
. فتح الحزمة
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72
Find the surface area of the portion of the sphere satisfying
satisfying فتح الحزمة
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73
Let where and . Let be a sector of angle in the sphere of radius , centered at the origin and oriented outward. Find .
where and . Let be a sector of angle in the sphere of radius , centered at the origin and oriented outward. Find . فتح الحزمة
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74
The area of the surface is which of the following?
A)
B)
C)
D)
E) None of the answers is correct.
is which of the following?A)
B)
C)
D)
E) None of the answers is correct.
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75
Find the flux of the field through the surface which is the part of the paraboloid where .
through the surface which is the part of the paraboloid where . فتح الحزمة
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76
Compute the surface integral where and is the surface of the sphere centered at the origin with radius 4.
where and is the surface of the sphere centered at the origin with radius 4. فتح الحزمة
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77
Find the surface area of the portion of the cone within the cylinder
within the cylinder فتح الحزمة
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78
Find the flux of the field through the surface oriented upward.
of the field through the surface oriented upward. فتح الحزمة
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79
Parametrize the surface first using Cartesian coordinates and then using cylindrical coordinates. Assume x and y take on all real values.
first using Cartesian coordinates and then using cylindrical coordinates. Assume x and y take on all real values. فتح الحزمة
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80
Find the area of the ellipse cut from the plane by the cylinder .
by the cylinder . فتح الحزمة
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