Question 1

Find a parametric representation for the part of the elliptic paraboloid   that lies in front of the plane x = 0.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 2

Evaluate   , where C is given by   Round your answer to two decimal place.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 3

Use Green's Theorem to find the work done by the force   in moving a particle in the positive direction once around the triangle with vertices   ,   , and   .

Accepted Answer

A)    
B) 42 
C)    
D) 7 
A)    
B) 42 
C)    
D) 7

Question 4

Use Green's Theorem to evaluate the line integral along the given positively oriented curve.   C is the ellipse

Accepted Answer

The answer of Use Green's Theorem to evaluate the line...

Question 5

Find the curl of   .

Accepted Answer

The answer of Find the curl of   ....

Question 6

Determine whether F is conservative. If so, find a function f such that   .

Accepted Answer

A)    
B) not conservative 
C)    
D)    
A)    
B) not conservative 
C)    
D)

Question 7

Let S be the cube with vertices   . Approximate   by using a Riemann sum as in Definition 1, taking the patches   to be the squares that are the faces of the cube and the points   to be the centers of the squares.

Accepted Answer

A)    
B)    
C)    
D)    
E) none of these 
A)    
B)    
C)    
D)    
E) none of these

Question 8

A thin wire in the shape of a quarter-circle   ,   , has a linear mass density   . Find the mass and the location of the center of mass of the wire.

Accepted Answer

The answer of A thin wire in the shape of...

Question 9

The temperature at the point   in a substance with conductivity   is   Find the rate of heat flow inward across the cylindrical

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 10

Let D be a region bounded by a simple closed path C in the xy. Then the coordinates of the centroid   where A is the area of D. Find the centroid of the triangle with vertices (0, 0), (   , 0) and (0,   ).

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 11

Use Stoke's theorem to calculate the surface integral   where   and S is the part of the cone

Accepted Answer

The answer of Use Stoke's theorem to calculate the surface...

Question 12

Find a function f such that   and use it to evaluate   along the given curve C.   C is the upper semicircle that starts at (1, 2) and ends at (5, 2).

Accepted Answer

The answer of Find a function f such that ...

Question 13

Find the gradient vector field of

Accepted Answer

The answer of Find the gradient vector field of ...

Question 14

Determine whether or not F is a conservative vector field. If it is, find a function f such that     .

Accepted Answer

The answer of Determine whether or not F is a...

Question 15

A plane lamina with constant density   occupies a region in the xy-plane bounded by a simple closed path C. Its moments of inertia about the axes are   Find the moments of inertia about the axes, if C is a rectangle with vertices (0, 0), (4, 0), (4, 5) and   .

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 16

Evaluate   .   ; S is the part of the cone   between the planes   and   .

Accepted Answer

A)    
B)    
C)      
D) 0 
A)    
B)    
C)      
D) 0

Question 17

Use Stokes' Theorem to evaluate   S consists of the top and the four sides (but not the bottom) of the cube with vertices   oriented outward.

Accepted Answer

The answer of Use Stokes' Theorem to evaluate  ...

Find a parametric representation for the part of the elliptic paraboloid that lies in front of the plane x = 0.

Evaluate , where C is given by Round your answer to two decimal place.

Use Green's Theorem to find the work done by the force in moving a particle in the positive direction once around the triangle with vertices , , and .

Use Green's Theorem to evaluate the line integral along the given positively oriented curve. C is the ellipse

Find the curl of .

Determine whether F is conservative. If so, find a function f such that .

Let S be the cube with vertices . Approximate by using a Riemann sum as in Definition 1, taking the patches to be the squares that are the faces of the cube and the points to be the centers of the squares.

A thin wire in the shape of a quarter-circle , , has a linear mass density . Find the mass and the location of the center of mass of the wire.

The temperature at the point in a substance with conductivity is Find the rate of heat flow inward across the cylindrical

Let D be a region bounded by a simple closed path C in the xy. Then the coordinates of the centroid where A is the area of D. Find the centroid of the triangle with vertices (0, 0), ( , 0) and (0, ).

Use Stoke's theorem to calculate the surface integral where and S is the part of the cone

Find a function f such that and use it to evaluate along the given curve C. C is the upper semicircle that starts at (1, 2) and ends at (5, 2).

Find the gradient vector field of

Determine whether or not F is a conservative vector field. If it is, find a function f such that .

A plane lamina with constant density occupies a region in the xy-plane bounded by a simple closed path C. Its moments of inertia about the axes are Find the moments of inertia about the axes, if C is a rectangle with vertices (0, 0), (4, 0), (4, 5) and .

Evaluate . ; S is the part of the cone between the planes and .

Use Stokes' Theorem to evaluate S consists of the top and the four sides (but not the bottom) of the cube with vertices oriented outward.

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Second-Order Differential Equations

Final Exam

Filters

Exam 16: Vector Calculus

Find a parametric representation for the part of the elliptic paraboloid that lies in front of the plane x = 0.

Evaluate , where C is given by Round your answer to two decimal place.

Use Green's Theorem to find the work done by the force in moving a particle in the positive direction once around the triangle with vertices , , and .

Use Green's Theorem to evaluate the line integral along the given positively oriented curve. C is the ellipse

Find the curl of .

Determine whether F is conservative. If so, find a function f such that .

Let S be the cube with vertices . Approximate by using a Riemann sum as in Definition 1, taking the patches to be the squares that are the faces of the cube and the points to be the centers of the squares.

A thin wire in the shape of a quarter-circle , , has a linear mass density . Find the mass and the location of the center of mass of the wire.

The temperature at the point in a substance with conductivity is Find the rate of heat flow inward across the cylindrical

Let D be a region bounded by a simple closed path C in the xy. Then the coordinates of the centroid where A is the area of D. Find the centroid of the triangle with vertices (0, 0), ( , 0) and (0, ).

Use Stoke's theorem to calculate the surface integral where and S is the part of the cone

Find a function f such that and use it to evaluate along the given curve C. C is the upper semicircle that starts at (1, 2) and ends at (5, 2).

Find the gradient vector field of

Determine whether or not F is a conservative vector field. If it is, find a function f such that .

A plane lamina with constant density occupies a region in the xy-plane bounded by a simple closed path C. Its moments of inertia about the axes are Find the moments of inertia about the axes, if C is a rectangle with vertices (0, 0), (4, 0), (4, 5) and .

Evaluate . ; S is the part of the cone between the planes and .

Use Stokes' Theorem to evaluate S consists of the top and the four sides (but not the bottom) of the cube with vertices oriented outward.

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Second-Order Differential Equations

Final Exam

Filters