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Deck 15: Vector Anal
Full screen (f)
Question
Match the following vector-valued function with its graph. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Evaluate 
, where 
and S is the closed surface of the solid bounded by the graphs, 
and 
, and the coordinate planes.
A)
B) 0
C)
D)
E)
, where and S is the closed surface of the solid bounded by the graphs, and , and the coordinate planes. A)
B) 0
C)
D)
E)
Question
Let 
and let S be the graph of 
. Verify Stokes's Theorem by evaluating 
as a line integral and as a double integral.
A)
B)
C) 0
D)
E)
and let S be the graph of . Verify Stokes's Theorem by evaluating as a line integral and as a double integral.
A)
B)
C) 0
D)
E)
Question
Use Stokes's Theorem to evaluate 
where 
and S is the first-octant portion of 
over 
. Use a computer algebra system to verify your result.
A)
B) 0
C)
D)
E)
where and S is the first-octant portion of over . Use a computer algebra system to verify your result. A)
B) 0
C)
D)
E)
Question
Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function 
and sketch the graph.
A)
B)
C)
D)
E)
and sketch the graph. A)
B)
C)
D)
E)
Question
Use Stokes's Theorem to evaluate 
. Use a computer algebra system to verify your results. Note: C is oriented counterclockwise as viewed from above. 
A)
B)
C)
D)
E)
. Use a computer algebra system to verify your results. Note: C is oriented counterclockwise as viewed from above. A)
B)
C)
D)
E)
Question
Match the following vector-valued function with its graph. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Use a computer algebra system and the result "The area of a plane region bounded by the simple closed path 
given in polar coordinates is 
" to find the area of the region bounded by the graphs of the polar equation 
.
A)
B)
C)
D)
E)
given in polar coordinates is " to find the area of the region bounded by the graphs of the polar equation . A)
B)
C)
D)
E)
Question
Let 
and let S be the surface bounded by 
and 
. Verify the Divergence Theorem by evaluating 
as a surface integral and as a triple integral. Round your answer to two decimal places.
A)
B)
C)
D)
E)
and let S be the surface bounded by and . Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. Round your answer to two decimal places.
A)
B)
C)
D)
E)
Question
Use Green's Theorem to evaluate the integral 
for the path 
defined as 
.
A)
B)
C)
D)
E)
for the path defined as . A)
B)
C)
D)
E)
Question
Set up and evaluate a line integral to find the area of the region R bounded by the graph of 
.
A)
where 
B)
where 
C)
where 
D)
where 
E)
where 
.A)
where B)
where C)
where D)
where E)
where Question
Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
Calculate the line integral along 
for 
and C is any path starting at the point 
and ending at 
.
A)
B)
C)
D)
E)
for and C is any path starting at the point and ending at . A)
B)
C)
D)
E)
Question
The surface of the dome on a new museum is given by 
, where 
and 
and 
is in meters. Find the surface area of the dome.
A)
B)
C)
D)
E)
, where and and is in meters. Find the surface area of the dome. A)
B)
C)
D)
E)
Question
Find the curl of the vector field 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
Sketch several representative vectors in the vector field given by 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
Find a vector-valued function for the hyperboloid 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
Find the work done by a person weighing 
pounds walking exactly one revolution up a circular helical staircase of radius 
feet if the person rises 
feet.
A)
B)
C)
D)
E)
pounds walking exactly one revolution up a circular helical staircase of radius feet if the person rises feet.A)
B)
C)
D)
E)
Question
Sketch the vector field 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
Use a computer algebra system and the result "The area of a plane region bounded by the simple closed path 
given in polar coordinates is 
" to find the area of the region bounded by the graphs of the polar equation 
. Round your answer to two decimal places.
A)
B)
C)
D)
E)
given in polar coordinates is " to find the area of the region bounded by the graphs of the polar equation . Round your answer to two decimal places. A)
B)
C)
D)
E)
Question
Find 
. 
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
Find a piecewise smooth parametrization of the path C given in the following graph. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Find the conservative vector field for the potential function 
by finding its gradient.
A)
B)
C)
D)
E)
by finding its gradient.A)
B)
C)
D)
E)
Question
Find the total mass of the wire with density 
. 
, 
, 
A)
B)
C)
D)
E)
. , , A)
B)
C)
D)
E)
Question
Determine whether the vector field is conservative. If it is, find a potential function for the vector field. 
A) conservative with potential function
B) conservative with potential function
C) conservative with potential function
D) conservative with potential function
E) not conservative
A) conservative with potential function
B) conservative with potential function
C) conservative with potential function
D) conservative with potential function
E) not conservative
Question
Find the value of the line integral 
where 
and 
.
where and . Question
Determine whether the vector field is conservative. If it is, find a potential function for the vector field. 
A) conservative with potential function
B) conservative with potential function
C) conservative with potential function
D) conservative with potential function
E) not conservative
A) conservative with potential function
B) conservative with potential function
C) conservative with potential function
D) conservative with potential function
E) not conservative
Question
Find a vector-valued function whose graph is the cylinder 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
Evaluate 
, where 
is the unit circle given by 
.
A)
B)
C)
D)
E)
, where is the unit circle given by . A)
B)
C)
D)
E)
Question
Use Divergence Theorem to evaluate 
and find the outward flux of 
through the surface S of the solid bounded by the sphere 
.
A)
B)
C)
D)
E)
and find the outward flux of through the surface S of the solid bounded by the sphere . A)
B)
C)
D)
E)
Question
Use Green's Theorem to calculate the work done by the force 
on a particle that is moving counterclockwise around the closed path 
where 
is the boundary of the region lying between the graphs of 
, and 
. Round your answer to two decimal places.
A)
B)
C)
D)
E)
on a particle that is moving counterclockwise around the closed path where is the boundary of the region lying between the graphs of , and . Round your answer to two decimal places. A)
B)
C)
D)
E)
Question
Find the gradient vector for the scalar function. (That is, find the conservative vector field for the potential function.) 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Use the Divergence Theorem to evaluate 
and find the outward flux of 
through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. 
A)
B)
C)
D)
E)
and find the outward flux of through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. A)
B)
C)
D)
E)
Question
Find the curl of the vector field 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
Find the moments of inertia for a wire that lies along 
, 
with density 
.
A)
B)
C)
D)
E)
, with density . A)
B)
C)
D)
E)
Question
Determine whether the vector field is conservative. 
A) conservative
B) not conservative
A) conservative
B) not conservative
Question
Determine the tangent plane for the hyperboloid 
at 
.
A)
B)
C)
D)
E)
at . A)
B)
C)
D)
E)
Question
Use Green's Theorem to evaluate the line integral 
where 
is 
.
A)
B)
C)
D)
E)
where is . A)
B)
C)
D)
E)
Question
Find the mass of the surface lamina S of density 
. 
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
A tractor engine has a steel component with a circular base modeled by the vector-valued function 
. Its height is given by 
. (All measurements of the component are given in centimeters.) Find the lateral surface area of the component. Round your answer to two decimal places.
A)
B)
C)
D)
E)
. Its height is given by . (All measurements of the component are given in centimeters.) Find the lateral surface area of the component. Round your answer to two decimal places. A)
B)
C)
D)
E)
Question
Find the flux 
of through S, 
, where 
is the upward unit normal vector to S. 
A)
B)
C)
D)
E)
of through S, , where is the upward unit normal vector to S. A)
B)
C)
D)
E)
Question
Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function 
about the x-axis.
A)
B)
C)
D)
E)
about the x-axis. A)
B)
C)
D)
E)
Question
Find an equation of the tangent plane to the surface represented by the vector-valued function at the given point. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Evaluate the integral 
along the path 
, defined as y-axis from 
to 
.
A)
B)
C)
D)
E)
along the path , defined as y-axis from to . A)
B)
C)
D)
E)
Question
Use Green's Theorem to calculate the work done by the force 
on a particle that is moving counterclockwise around the closed path 
. 
A)
B)
C)
D)
E)
on a particle that is moving counterclockwise around the closed path . A)
B)
C)
D)
E)
Question
Find the divergence at 
for the vector field 
.
A)
B)
C)
D)
E) 0
for the vector field .A)
B)
C)
D)
E) 0
Question
Find the divergence of the vector field. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Let 
and let C be the triangle with vertices of 
oriented counterclockwise. Use Stokes's Theorem to evaluate 
.
A) 0
B)
C)
D)
E)
and let C be the triangle with vertices of oriented counterclockwise. Use Stokes's Theorem to evaluate . A) 0
B)
C)
D)
E)
Question
Use Stokes's Theorem to evaluate 
where 
and S is 
over 
in the first octant. Use a computer algebra system to verify your result.
A) 0
B)
C)
D)
E)
where and S is over in the first octant. Use a computer algebra system to verify your result. A) 0
B)
C)
D)
E)
Question
Evaluate 
where 
is represented by 
. 
, 
A)
B) 0
C)
D)
E)
where is represented by . , A)
B) 0
C)
D)
E)
Question
Question
For the vector field 
, find the value of 
for which the field is conservative.
A)
B)
C)
D)
E)
is not conservative for any value of 
.
, find the value of for which the field is conservative. A)
B)
C)
D)
E)
is not conservative for any value of . Question
Use a computer algebra system to evaluate 
where S is 
. Round your answer to two decimal places.
A) 4,798.52
B) 10.80
C) 2,399.26
D) 20,349.51
E) 3,280.50
where S is . Round your answer to two decimal places. A) 4,798.52
B) 10.80
C) 2,399.26
D) 20,349.51
E) 3,280.50
Question
Let 
and let S be the graph of 
oriented counterclockwise. Use Stokes's Theorem to evaluate 
.
A) 0
B)
C)
D)
E)
and let S be the graph of oriented counterclockwise. Use Stokes's Theorem to evaluate . A) 0
B)
C)
D)
E)
Question
Let 
be an electrostatic field. Use Gauss's Law to find the total charge enclosed by the closed surface consisting of the hemisphere 
and its circular base in the xy-plane.
A)
B)
C)
D)
E)
be an electrostatic field. Use Gauss's Law to find the total charge enclosed by the closed surface consisting of the hemisphere and its circular base in the xy-plane. A)
B)
C)
D)
E)
Question
Evaluate the line integral 
using the Fundamental Theorem of Line Integrals, where C is the line segment from 
to 
.
A)
B)
C)
D)
E)
using the Fundamental Theorem of Line Integrals, where C is the line segment from to . A)
B)
C)
D)
E)
Question
Evaluate 
along the path C, defined as y-axis from 
to 
.
A)
B)
C)
D)
E)
along the path C, defined as y-axis from to .A)
B)
C)
D)
E)
Question
Use Stokes's Theorem to evaluate 
. Use a computer algebra system to verify your results. Note: C is oriented counterclockwise as viewed from above. 
C: triangle with vertices
A)
B)
C)
D)
E)
. Use a computer algebra system to verify your results. Note: C is oriented counterclockwise as viewed from above. C: triangle with vertices
A)
B)
C)
D)
E)
Question
Use Stokes's Theorem to evaluate 
where 
and S is the first-octant portion of 
over 
. Use a computer algebra system to verify your result.
A)
B)
C)
D) 0
E)
where and S is the first-octant portion of over . Use a computer algebra system to verify your result. A)
B)
C)
D) 0
E)
Question
Determine whether or not the vector field is conservative. 
A) conservative
B) not conservative
A) conservative
B) not conservative
Question
Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. 
C: circle
clockwise from 
to 
A)
B)
C)
D)
E)
C: circle
clockwise from to A)
B)
C)
D)
E)
Question
Use Divergence Theorem to evaluate 
and find the outward flux of 
through the surface S of the solid bounded by the planes 
and 
.
A)
B)
C)
D)
E)
and find the outward flux of through the surface S of the solid bounded by the planes and . A)
B)
C)
D)
E)
Question
Calculate the line integral along 
for 
and C is any path starting at the point 
and ending at 
.
A)
B)
C)
D)
E)
for and C is any path starting at the point and ending at . A)
B)
C)
D)
E)
Question
Find the value of the line integral 
where 
is an ellipse 
from 
to 
.
where is an ellipse from to . Question
Evaluate 
where S is the closed surface of the solid bounded by the graphs of 
and 
. 
A)
B)
C) 0
D)
E)
where S is the closed surface of the solid bounded by the graphs of and . A)
B)
C) 0
D)
E)
Question
Find 
. 
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
Find the curl of the vector field 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Compute 
for the vector field given by 
.
A)
B)
C)
D)
E)
for the vector field given by .A)
B)
C)
D)
E)
Question
Find the work done by the force field 
in moving an object from P to Q. 
A) 4,664
B) 9,329
C) 13,994
D) 6,997
E) 11,662
in moving an object from P to Q. A) 4,664
B) 9,329
C) 13,994
D) 6,997
E) 11,662
Question
Find the area of the surface over the given region. Use a computer algebra system to verify your results.
The part of the cone,
Where
and 
.
A)
B)
C)
D)
E)
The part of the cone,
Where
and .
A)
B)
C)
D)
E)
Question
Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. 
C: a smooth curve from
to 
.
A)
B)
C)
D)
E)
C: a smooth curve from
to .
A)
B)
C)
D)
E)
Question
Find the divergence of the vector field at the given point. 
, 
A)
B)
C)
D)
E)
, A)
B)
C)
D)
E)
Question
Evaluate 
, where 
.
A)
B)
C)
D) 0
E)
, where . A)
B)
C)
D) 0
E)
Question
Evaluate 
, where 
is 
.
A)
B)
C)
D)
E)
, where is . A)
B)
C)
D)
E)
Question
Find the area of the surface given by 
, where 
and 
.
A)
B)
C)
D)
E)
, where and . A)
B)
C)
D)
E)
Question
Find a vector-valued function whose graph is the ellipsoid 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
Use Green's Theorem to evaluate the integral 
For the path C: boundary of the region lying between the graphs of
and 
.
A)
B)
C)
D)
E)
For the path C: boundary of the region lying between the graphs of
and .
A)
B)
C)
D)
E)
Question
Find the work done by the force field 
on a particle moving along the given path. 
, 
from 
to 
.
A)
B)
C)
D)
E)
on a particle moving along the given path. , from to .
A)
B)
C)
D)
E)
Question
Determine whether the vector field is conservative. If it is, find a potential function for the vector field. 
A) conservative with potential function
B) conservative with potential function
C) conservative with potential function
D) conservative with potential function
E) not conservative
A) conservative with potential function
B) conservative with potential function
C) conservative with potential function
D) conservative with potential function
E) not conservative
Question
Use the Divergence Theorem to evaluate 
and find the outward flux of 
through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. 
A)
B)
C)
D)
E) 0
and find the outward flux of through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. A)
B)
C)
D)
E) 0
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Full screen (f)
Deck 15: Vector Anal
1
Match the following vector-valued function with its graph.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
B
2
Evaluate , where and S is the closed surface of the solid bounded by the graphs, and , and the coordinate planes.
A)
B) 0
C)
D)
E)
, where and S is the closed surface of the solid bounded by the graphs, and , and the coordinate planes. A)
B) 0
C)
D)
E)
B
3
Let and let S be the graph of . Verify Stokes's Theorem by evaluating as a line integral and as a double integral.
A)
B)
C) 0
D)
E)
and let S be the graph of . Verify Stokes's Theorem by evaluating as a line integral and as a double integral.
A)
B)
C) 0
D)
E)
D
4
Use Stokes's Theorem to evaluate where and S is the first-octant portion of over . Use a computer algebra system to verify your result.
A)
B) 0
C)
D)
E)
where and S is the first-octant portion of over . Use a computer algebra system to verify your result. A)
B) 0
C)
D)
E)
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5
Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function and sketch the graph.
A)
B)
C)
D)
E)
and sketch the graph. A)
B)
C)
D)
E)
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6
Use Stokes's Theorem to evaluate . Use a computer algebra system to verify your results. Note: C is oriented counterclockwise as viewed from above.
A)
B)
C)
D)
E)
. Use a computer algebra system to verify your results. Note: C is oriented counterclockwise as viewed from above. A)
B)
C)
D)
E)
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7
Match the following vector-valued function with its graph.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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8
Use a computer algebra system and the result "The area of a plane region bounded by the simple closed path given in polar coordinates is " to find the area of the region bounded by the graphs of the polar equation .
A)
B)
C)
D)
E)
given in polar coordinates is " to find the area of the region bounded by the graphs of the polar equation . A)
B)
C)
D)
E)
Unlock Deck
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9
Let and let S be the surface bounded by and . Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. Round your answer to two decimal places.
A)
B)
C)
D)
E)
and let S be the surface bounded by and . Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. Round your answer to two decimal places.
A)
B)
C)
D)
E)
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10
Use Green's Theorem to evaluate the integral for the path defined as .
A)
B)
C)
D)
E)
for the path defined as . A)
B)
C)
D)
E)
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11
Set up and evaluate a line integral to find the area of the region R bounded by the graph of .
A) where
B) where
C) where
D) where
E) where
.A)
where B)
where C)
where D)
where E)
where Unlock Deck
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12
Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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13
Calculate the line integral along for and C is any path starting at the point and ending at .
A)
B)
C)
D)
E)
for and C is any path starting at the point and ending at . A)
B)
C)
D)
E)
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14
The surface of the dome on a new museum is given by , where and and is in meters. Find the surface area of the dome.
A)
B)
C)
D)
E)
, where and and is in meters. Find the surface area of the dome. A)
B)
C)
D)
E)
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15
Find the curl of the vector field .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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16
Sketch several representative vectors in the vector field given by .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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17
Find a vector-valued function for the hyperboloid .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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18
Find the work done by a person weighing pounds walking exactly one revolution up a circular helical staircase of radius feet if the person rises feet.
A)
B)
C)
D)
E)
pounds walking exactly one revolution up a circular helical staircase of radius feet if the person rises feet.A)
B)
C)
D)
E)
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19
Sketch the vector field .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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20
Use a computer algebra system and the result "The area of a plane region bounded by the simple closed path given in polar coordinates is " to find the area of the region bounded by the graphs of the polar equation . Round your answer to two decimal places.
A)
B)
C)
D)
E)
given in polar coordinates is " to find the area of the region bounded by the graphs of the polar equation . Round your answer to two decimal places. A)
B)
C)
D)
E)
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21
Find .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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22
Find a piecewise smooth parametrization of the path C given in the following graph.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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23
Find the conservative vector field for the potential function by finding its gradient.
A)
B)
C)
D)
E)
by finding its gradient.A)
B)
C)
D)
E)
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24
Find the total mass of the wire with density . , ,
A)
B)
C)
D)
E)
. , , A)
B)
C)
D)
E)
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25
Determine whether the vector field is conservative. If it is, find a potential function for the vector field.
A) conservative with potential function
B) conservative with potential function
C) conservative with potential function
D) conservative with potential function
E) not conservative
A) conservative with potential function
B) conservative with potential function
C) conservative with potential function
D) conservative with potential function
E) not conservative
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26
Find the value of the line integral where and .
where and . Unlock Deck
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27
Determine whether the vector field is conservative. If it is, find a potential function for the vector field.
A) conservative with potential function
B) conservative with potential function
C) conservative with potential function
D) conservative with potential function
E) not conservative
A) conservative with potential function
B) conservative with potential function
C) conservative with potential function
D) conservative with potential function
E) not conservative
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28
Find a vector-valued function whose graph is the cylinder .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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29
Evaluate , where is the unit circle given by .
A)
B)
C)
D)
E)
, where is the unit circle given by . A)
B)
C)
D)
E)
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30
Use Divergence Theorem to evaluate and find the outward flux of through the surface S of the solid bounded by the sphere .
A)
B)
C)
D)
E)
and find the outward flux of through the surface S of the solid bounded by the sphere . A)
B)
C)
D)
E)
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31
Use Green's Theorem to calculate the work done by the force on a particle that is moving counterclockwise around the closed path where is the boundary of the region lying between the graphs of , and . Round your answer to two decimal places.
A)
B)
C)
D)
E)
on a particle that is moving counterclockwise around the closed path where is the boundary of the region lying between the graphs of , and . Round your answer to two decimal places. A)
B)
C)
D)
E)
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32
Find the gradient vector for the scalar function. (That is, find the conservative vector field for the potential function.)
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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33
Use the Divergence Theorem to evaluate and find the outward flux of through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.
A)
B)
C)
D)
E)
and find the outward flux of through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. A)
B)
C)
D)
E)
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34
Find the curl of the vector field .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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35
Find the moments of inertia for a wire that lies along , with density .
A)
B)
C)
D)
E)
, with density . A)
B)
C)
D)
E)
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36
Determine whether the vector field is conservative.
A) conservative
B) not conservative
A) conservative
B) not conservative
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37
Determine the tangent plane for the hyperboloid at .
A)
B)
C)
D)
E)
at . A)
B)
C)
D)
E)
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38
Use Green's Theorem to evaluate the line integral where is .
A)
B)
C)
D)
E)
where is . A)
B)
C)
D)
E)
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39
Find the mass of the surface lamina S of density .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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40
A tractor engine has a steel component with a circular base modeled by the vector-valued function . Its height is given by . (All measurements of the component are given in centimeters.) Find the lateral surface area of the component. Round your answer to two decimal places.
A)
B)
C)
D)
E)
. Its height is given by . (All measurements of the component are given in centimeters.) Find the lateral surface area of the component. Round your answer to two decimal places. A)
B)
C)
D)
E)
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41
Find the flux of through S, , where is the upward unit normal vector to S.
A)
B)
C)
D)
E)
of through S, , where is the upward unit normal vector to S. A)
B)
C)
D)
E)
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42
Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the x-axis.
A)
B)
C)
D)
E)
about the x-axis. A)
B)
C)
D)
E)
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43
Find an equation of the tangent plane to the surface represented by the vector-valued function at the given point.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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44
Evaluate the integral along the path , defined as y-axis from to .
A)
B)
C)
D)
E)
along the path , defined as y-axis from to . A)
B)
C)
D)
E)
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45
Use Green's Theorem to calculate the work done by the force on a particle that is moving counterclockwise around the closed path .
A)
B)
C)
D)
E)
on a particle that is moving counterclockwise around the closed path . A)
B)
C)
D)
E)
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46
Find the divergence at for the vector field .
A)
B)
C)
D)
E) 0
for the vector field .A)
B)
C)
D)
E) 0
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47
Find the divergence of the vector field.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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48
Let and let C be the triangle with vertices of oriented counterclockwise. Use Stokes's Theorem to evaluate .
A) 0
B)
C)
D)
E)
and let C be the triangle with vertices of oriented counterclockwise. Use Stokes's Theorem to evaluate . A) 0
B)
C)
D)
E)
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49
Use Stokes's Theorem to evaluate where and S is over in the first octant. Use a computer algebra system to verify your result.
A) 0
B)
C)
D)
E)
where and S is over in the first octant. Use a computer algebra system to verify your result. A) 0
B)
C)
D)
E)
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50
Evaluate where is represented by . ,
A)
B) 0
C)
D)
E)
where is represented by . , A)
B) 0
C)
D)
E)
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51
A stone weighing 5 pounds is attached to the end of a five-foot string and is whirled horizontally with one end held fixed. It makes 1 revolution per second. Find the work done by the force F that keeps the stone moving in a circular path.
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52
For the vector field , find the value of for which the field is conservative.
A)
B)
C)
D)
E) is not conservative for any value of .
, find the value of for which the field is conservative. A)
B)
C)
D)
E)
is not conservative for any value of . Unlock Deck
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53
Use a computer algebra system to evaluate where S is . Round your answer to two decimal places.
A) 4,798.52
B) 10.80
C) 2,399.26
D) 20,349.51
E) 3,280.50
where S is . Round your answer to two decimal places. A) 4,798.52
B) 10.80
C) 2,399.26
D) 20,349.51
E) 3,280.50
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54
Let and let S be the graph of oriented counterclockwise. Use Stokes's Theorem to evaluate .
A) 0
B)
C)
D)
E)
and let S be the graph of oriented counterclockwise. Use Stokes's Theorem to evaluate . A) 0
B)
C)
D)
E)
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55
Let be an electrostatic field. Use Gauss's Law to find the total charge enclosed by the closed surface consisting of the hemisphere and its circular base in the xy-plane.
A)
B)
C)
D)
E)
be an electrostatic field. Use Gauss's Law to find the total charge enclosed by the closed surface consisting of the hemisphere and its circular base in the xy-plane. A)
B)
C)
D)
E)
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56
Evaluate the line integral using the Fundamental Theorem of Line Integrals, where C is the line segment from to .
A)
B)
C)
D)
E)
using the Fundamental Theorem of Line Integrals, where C is the line segment from to . A)
B)
C)
D)
E)
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57
Evaluate along the path C, defined as y-axis from to .
A)
B)
C)
D)
E)
along the path C, defined as y-axis from to .A)
B)
C)
D)
E)
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58
Use Stokes's Theorem to evaluate . Use a computer algebra system to verify your results. Note: C is oriented counterclockwise as viewed from above.
C: triangle with vertices
A)
B)
C)
D)
E)
. Use a computer algebra system to verify your results. Note: C is oriented counterclockwise as viewed from above. C: triangle with vertices
A)
B)
C)
D)
E)
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59
Use Stokes's Theorem to evaluate where and S is the first-octant portion of over . Use a computer algebra system to verify your result.
A)
B)
C)
D) 0
E)
where and S is the first-octant portion of over . Use a computer algebra system to verify your result. A)
B)
C)
D) 0
E)
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60
Determine whether or not the vector field is conservative.
A) conservative
B) not conservative
A) conservative
B) not conservative
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61
Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results.
C: circle clockwise from to
A)
B)
C)
D)
E)
C: circle
clockwise from to A)
B)
C)
D)
E)
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62
Use Divergence Theorem to evaluate and find the outward flux of through the surface S of the solid bounded by the planes and .
A)
B)
C)
D)
E)
and find the outward flux of through the surface S of the solid bounded by the planes and . A)
B)
C)
D)
E)
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63
Calculate the line integral along for and C is any path starting at the point and ending at .
A)
B)
C)
D)
E)
for and C is any path starting at the point and ending at . A)
B)
C)
D)
E)
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64
Find the value of the line integral where is an ellipse from to .
where is an ellipse from to . Unlock Deck
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65
Evaluate where S is the closed surface of the solid bounded by the graphs of and .
A)
B)
C) 0
D)
E)
where S is the closed surface of the solid bounded by the graphs of and . A)
B)
C) 0
D)
E)
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66
Find .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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67
Find the curl of the vector field
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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68
Compute for the vector field given by .
A)
B)
C)
D)
E)
for the vector field given by .A)
B)
C)
D)
E)
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69
Find the work done by the force field in moving an object from P to Q.
A) 4,664
B) 9,329
C) 13,994
D) 6,997
E) 11,662
in moving an object from P to Q. A) 4,664
B) 9,329
C) 13,994
D) 6,997
E) 11,662
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70
Find the area of the surface over the given region. Use a computer algebra system to verify your results.
The part of the cone,
Where and .
A)
B)
C)
D)
E)
The part of the cone,
Where
and .
A)
B)
C)
D)
E)
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71
Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results.
C: a smooth curve from to .
A)
B)
C)
D)
E)
C: a smooth curve from
to .
A)
B)
C)
D)
E)
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72
Find the divergence of the vector field at the given point. ,
A)
B)
C)
D)
E)
, A)
B)
C)
D)
E)
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73
Evaluate , where .
A)
B)
C)
D) 0
E)
, where . A)
B)
C)
D) 0
E)
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74
Evaluate , where is .
A)
B)
C)
D)
E)
, where is . A)
B)
C)
D)
E)
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75
Find the area of the surface given by , where and .
A)
B)
C)
D)
E)
, where and . A)
B)
C)
D)
E)
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76
Find a vector-valued function whose graph is the ellipsoid .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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77
Use Green's Theorem to evaluate the integral
For the path C: boundary of the region lying between the graphs of and .
A)
B)
C)
D)
E)
For the path C: boundary of the region lying between the graphs of
and .
A)
B)
C)
D)
E)
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78
Find the work done by the force field on a particle moving along the given path. , from to .
A)
B)
C)
D)
E)
on a particle moving along the given path. , from to .
A)
B)
C)
D)
E)
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79
Determine whether the vector field is conservative. If it is, find a potential function for the vector field.
A) conservative with potential function
B) conservative with potential function
C) conservative with potential function
D) conservative with potential function
E) not conservative
A) conservative with potential function
B) conservative with potential function
C) conservative with potential function
D) conservative with potential function
E) not conservative
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80
Use the Divergence Theorem to evaluate and find the outward flux of through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.
A)
B)
C)
D)
E) 0
and find the outward flux of through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. A)
B)
C)
D)
E) 0
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Unlock Deck
Unlock for access to all 142 flashcards in this deck.
