Question 1

The equation of the tangent plane to x = u, y = v,   where u = 1 and v = 0 is

Accepted Answer

A)  x + 1 - 2y - z = 0 
B)  x + 1 + 2y + z = 0 
C)  x + 1 + 2y - z = 0 
D)  x - 2y + z = 0 
E)  x - z = 0 
A)  x + 1 - 2y - z = 0 
B)  x + 1 + 2y + z = 0 
C)  x + 1 + 2y - z = 0 
D)  x - 2y + z = 0 
E)  x - z = 0 E

Question 2

Find the volume of the solid bounded by   and the rectangle R = [0, 3] *[0, 3].

Accepted Answer

A)  9 cubic units 
B)  18 cubic units 
C)  6 cubic units 
D)  81 cubic units 
E)  54 cubic units 
A)  9 cubic units 
B)  18 cubic units 
C)  6 cubic units 
D)  81 cubic units 
E)  54 cubic units E

Question 3

Find the Jacobian if u = 2xy and v = 2x + 6.

Accepted Answer

A)    
B)    
C)    
D)    
E)  u + v 
A)    
B)    
C)    
D)    
E)  u + v D

Question 4

Use a double integral in polar coordinates to find the volume of the solid enclosed by x2 + y2 = 40 - z and z = 4.

Accepted Answer

Question 5

Find the mass of the tetrahedron in the first octant enclosed by the coordinate planes and the plane x + y + z = 1 if its density is given by  $\delta$(x, y, z) = 11xy.

Accepted Answer

The answer of Find the mass of the tetrahedron in...

Question 6

Evaluate

Accepted Answer

The answer of Evaluate  ...

Question 7

Use the theorem of Pappas to find the volume of the solid generated when the region enclosed by y = 3x2 and y = 3(8 - x2) is revolved about the line y = -2.
Obtain the centroid by symmetry.

Accepted Answer

Question 8

Evaluate

Accepted Answer

The answer of Evaluate  ...

Question 9

Find the Jacobian,   ; x = 4u + 8 + v, y = 3u - 5v.

Accepted Answer

The answer of Find the Jacobian,   ; x...

Question 10

Evaluate   by converting to an equivalent integral in polar coordinates. Sketch the region R.

Accepted Answer

The answer of Evaluate   by converting to an...

Question 11

Find   where R is the region in the first quadrant enclosed between y = x and y = x5.

Accepted Answer

A)    
B)  1 
C)    
D)    
E)  0 
A)    
B)  1 
C)    
D)    
E)  0

Question 12

Use a double integral in polar coordinates to find the volume that is inside the sphere x2 + y2 + z2 = 16, outside the cylinder x2 + y2 = 4 and above z = 0.

Accepted Answer

Question 13

Find the volume of the region bounded above by the plane   in the first octant.

Accepted Answer

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

Question 14

A lamina with density  $\delta$(x, y) = 2x2 + y2 + 9 is bounded by x = y, x = 0, y = 0, y = 2. Find its center of mass.

Accepted Answer

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

Question 15

Find the volume of the solid in the first octant enclosed by z = 3(4 - y2), z = 0, x = 0, y = x, and y = 2.

Accepted Answer

Question 16

Find the volume of the solid enclosed by y = x2 - x + a, y = x + a, z = x + 1, and the xy-plane.

Accepted Answer

Question 17

Use an appropriate transform to evaluate   where R is the region enclosed by   ,   , and   .

Accepted Answer

The answer of Use an appropriate transform to evaluate ...

Question 18

Use a triple integral to find the volume of the solid in the first octant enclosed by the cylinder x = 4 - y2 and the planes z = y, x = 0, and z = 0.

Accepted Answer

Question 19

A lamina with density $\delta$ (x, y) = 2x2 + y2 is bounded by x = y, x = 0, y = 0, y = 2. Find its moment of inertia about the y-axis.

Accepted Answer

A)    
B)  16 
C)    
D)    
E)    
A)    
B)  16 
C)    
D)    
E)

Question 20

Use a CAS to solve the problem.

Accepted Answer

The answer of Use a CAS to solve the problem....

The equation of the tangent plane to x = u, y = v, where u = 1 and v = 0 is

Find the volume of the solid bounded by and the rectangle R = [0, 3] *[0, 3].

Find the Jacobian if u = 2xy and v = 2x + 6.

Use a double integral in polar coordinates to find the volume of the solid enclosed by x² + y² = 40 - z and z = 4.

Find the mass of the tetrahedron in the first octant enclosed by the coordinate planes and the plane x + y + z = 1 if its density is given by $\delta$ (x, y, z) = 11xy.

Evaluate

Use the theorem of Pappas to find the volume of the solid generated when the region enclosed by y = 3x² and y = 3(8 - x²) is revolved about the line y = -2. Obtain the centroid by symmetry.

Evaluate

Find the Jacobian, ; x = 4u + 8 + v, y = 3u - 5v.

Evaluate by converting to an equivalent integral in polar coordinates. Sketch the region R.

Find where R is the region in the first quadrant enclosed between y = x and y = x⁵.

Use a double integral in polar coordinates to find the volume that is inside the sphere x² + y² + z² = 16, outside the cylinder x² + y² = 4 and above z = 0.

Find the volume of the region bounded above by the plane in the first octant.

A lamina with density $\delta$ (x, y) = 2x² + y² + 9 is bounded by x = y, x = 0, y = 0, y = 2. Find its center of mass.

Find the volume of the solid in the first octant enclosed by z = 3(4 - y²), z = 0, x = 0, y = x, and y = 2.

Find the volume of the solid enclosed by y = x² - x + a, y = x + a, z = x + 1, and the xy-plane.

Use an appropriate transform to evaluate where R is the region enclosed by , , and .

Use a triple integral to find the volume of the solid in the first octant enclosed by the cylinder x = 4 - y² and the planes z = y, x = 0, and z = 0.

A lamina with density $\delta$ (x, y) = 2x² + y² is bounded by x = y, x = 0, y = 0, y = 2. Find its moment of inertia about the y-axis.

Use a CAS to solve the problem.

Limits and Continuity

The Derivative

Topics in Deifferentiation

The Derivative in Graphing and Applications

Integration

Applications of the Definite Integral in Geometry, Science and Engineering

Principle of Integral Evaluation

Mathematical Modeling With Differential Equations

Infinte Series

Parametric and Polar Curves; Conic Sections

Three-Dimensional Space; Vectors

Vector-Valued Functions

Partial Derivatives

Topics in Vector Calculus

Filters

Exam 14: Multiple Integrals

The equation of the tangent plane to x = u, y = v, where u = 1 and v = 0 is

Find the volume of the solid bounded by and the rectangle R = [0, 3] *[0, 3].

Find the Jacobian if u = 2xy and v = 2x + 6.

Use a double integral in polar coordinates to find the volume of the solid enclosed by x2 + y2 = 40 - z and z = 4.

Find the mass of the tetrahedron in the first octant enclosed by the coordinate planes and the plane x + y + z = 1 if its density is given by δ\deltaδ (x, y, z) = 11xy.

Evaluate

Use the theorem of Pappas to find the volume of the solid generated when the region enclosed by y = 3x2 and y = 3(8 - x2) is revolved about the line y = -2. Obtain the centroid by symmetry.

Evaluate

Find the Jacobian, ; x = 4u + 8 + v, y = 3u - 5v.

Evaluate by converting to an equivalent integral in polar coordinates. Sketch the region R.

Find where R is the region in the first quadrant enclosed between y = x and y = x5.

Use a double integral in polar coordinates to find the volume that is inside the sphere x2 + y2 + z2 = 16, outside the cylinder x2 + y2 = 4 and above z = 0.

Find the volume of the region bounded above by the plane in the first octant.

A lamina with density δ\deltaδ (x, y) = 2x2 + y2 + 9 is bounded by x = y, x = 0, y = 0, y = 2. Find its center of mass.

Find the volume of the solid in the first octant enclosed by z = 3(4 - y2), z = 0, x = 0, y = x, and y = 2.

Find the volume of the solid enclosed by y = x2 - x + a, y = x + a, z = x + 1, and the xy-plane.

Use an appropriate transform to evaluate where R is the region enclosed by , , and .

Use a triple integral to find the volume of the solid in the first octant enclosed by the cylinder x = 4 - y2 and the planes z = y, x = 0, and z = 0.

A lamina with density δ\deltaδ (x, y) = 2x2 + y2 is bounded by x = y, x = 0, y = 0, y = 2. Find its moment of inertia about the y-axis.

Use a CAS to solve the problem.

Limits and Continuity

The Derivative

Topics in Deifferentiation

The Derivative in Graphing and Applications

Integration

Applications of the Definite Integral in Geometry, Science and Engineering

Principle of Integral Evaluation

Mathematical Modeling With Differential Equations

Infinte Series

Parametric and Polar Curves; Conic Sections

Three-Dimensional Space; Vectors

Vector-Valued Functions

Partial Derivatives

Topics in Vector Calculus

Filters

Use a double integral in polar coordinates to find the volume of the solid enclosed by x² + y² = 40 - z and z = 4.

Find the mass of the tetrahedron in the first octant enclosed by the coordinate planes and the plane x + y + z = 1 if its density is given by $\delta$ (x, y, z) = 11xy.

Use the theorem of Pappas to find the volume of the solid generated when the region enclosed by y = 3x² and y = 3(8 - x²) is revolved about the line y = -2. Obtain the centroid by symmetry.

Find where R is the region in the first quadrant enclosed between y = x and y = x⁵.

Use a double integral in polar coordinates to find the volume that is inside the sphere x² + y² + z² = 16, outside the cylinder x² + y² = 4 and above z = 0.

A lamina with density $\delta$ (x, y) = 2x² + y² + 9 is bounded by x = y, x = 0, y = 0, y = 2. Find its center of mass.

Find the volume of the solid in the first octant enclosed by z = 3(4 - y²), z = 0, x = 0, y = x, and y = 2.

Find the volume of the solid enclosed by y = x² - x + a, y = x + a, z = x + 1, and the xy-plane.

Use a triple integral to find the volume of the solid in the first octant enclosed by the cylinder x = 4 - y² and the planes z = y, x = 0, and z = 0.

A lamina with density $\delta$ (x, y) = 2x² + y² is bounded by x = y, x = 0, y = 0, y = 2. Find its moment of inertia about the y-axis.