Question 1

A lamina with density  $\delta$(x, y) = 2xy + 11 is bounded by x = 2, x = 0, y = 0, y = x. Find its center of mass.

Accepted Answer

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

Question 2

Find the volume of the solid formed by the right hemisphere of   .

Accepted Answer

A)    
B)    
C)    
D)    
E)   $\pi$ 
A)    
B)    
C)    
D)    
E)   $\pi$

Question 3

Find the volume of the solid formed by the right hemisphere of   .

Accepted Answer

A)    
B)    
C)    
D)    
E)   $\pi$ 
A)    
B)    
C)    
D)    
E)   $\pi$

Question 4

A uniform beam 1 m in length is supported at its center by a fulcrum. A mass of 20kg is placed at the left end, a mass of 8kg is placed on the beam 10 m from the left end, and a third mass is placed 4 m from the right end. What mass should the third mass be to achieve equilibrium?

Accepted Answer

A)  28kg 
B)  36kg 
C)  16kg 
D)  20kg 
E)  10kg 
A)  28kg 
B)  36kg 
C)  16kg 
D)  20kg 
E)  10kg

Question 5

The vector normal to the surface given by   ,   , and   when   and   is

Accepted Answer

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

Question 6

Use cylindrical coordinates to evaluate   , where R is the solid enclosed by   and   .

Accepted Answer

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

Question 7

Compute   :

Accepted Answer

1.981683 *

Question 8

Find a parametric representation of the surface in terms of the parameters r and $\theta$, where (r, $\theta$, z) are the cylindrical coordinates of a point on the surface   .

Accepted Answer

x = r cos

Question 9

Find the volume of the region given by   lying above the xy-plane.

Accepted Answer

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

Question 10

Use a CAS to solve the problem.

Accepted Answer

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

Question 11

Evaluate the double integral   where R is the rectangular region bounded by the lines x = *1, x = 2, y = 0, and y = 4.

Accepted Answer

The answer of Evaluate the double integral   where...

Question 12

The cylindrical parameterization of   is

Accepted Answer

A)  x = r cos $\theta$, y = r sin $\theta$, z = e r 
B)  x = r sin $\theta$, y = r cos $\theta$, z = e r 
C)  x = r cos $\theta$, y = r sin$\theta$,   
D)  x = r sin $\theta$, y = r cos $\theta$, z = re r sin $\theta$ 
E)  x = r, y = r, z = re r 
A)  x = r cos $\theta$, y = r sin $\theta$, z = e r 
B)  x = r sin $\theta$, y = r cos $\theta$, z = e r 
C)  x = r cos $\theta$, y = r sin$\theta$,   
D)  x = r sin $\theta$, y = r cos $\theta$, z = re r sin $\theta$ 
E)  x = r, y = r, z = re r

Question 13

Use a triple integral to find the volume of the solid enclosed by x2 = 4y, y + z = 1, and z = 0.

Accepted Answer

Question 14

The equation of the tangent plane to x = u, y = v, z = u + v2 where u = 2 and v = 2 is

Accepted Answer

A)  x - 2 + 2(y - 2) + z - 6 = 0 
B)  x - 2 + 4(y - 2) - z + 6 = 0 
C)  x - 2 + 2y - 2 + z + 6 = 0 
D)  x - 2 + 2y - 4 - z + 6 = 0 
E)  x + 2 + 2y - 4 - z + 6 = 0 
A)  x - 2 + 2(y - 2) + z - 6 = 0 
B)  x - 2 + 4(y - 2) - z + 6 = 0 
C)  x - 2 + 2y - 2 + z + 6 = 0 
D)  x - 2 + 2y - 4 - z + 6 = 0 
E)  x + 2 + 2y - 4 - z + 6 = 0

Question 15

Evaluate   , by first sketching R then reversing the order of integration.

Accepted Answer

The answer of Evaluate   , by first sketching...

Question 16

Evaluate the double integral   where R is the rectangle bounded by -1$\le$ x $\le$ 3 and 0 $\le$y $\le$ 3.

Accepted Answer

The answer of Evaluate the double integral   where...

Question 17

Use polar coordinates to evaluate   where R is the region enclosed by   and x $\ge$ 0.

Accepted Answer

The answer of Use polar coordinates to evaluate  ...

A lamina with density $\delta$ (x, y) = 2xy + 11 is bounded by x = 2, x = 0, y = 0, y = x. Find its center of mass.

Find the volume of the solid formed by the right hemisphere of .

Find the volume of the solid formed by the right hemisphere of .

A uniform beam 1 m in length is supported at its center by a fulcrum. A mass of 20kg is placed at the left end, a mass of 8kg is placed on the beam 10 m from the left end, and a third mass is placed 4 m from the right end. What mass should the third mass be to achieve equilibrium?

The vector normal to the surface given by , , and when and is

Use cylindrical coordinates to evaluate , where R is the solid enclosed by and .

Compute :

Find the volume of the region given by lying above the xy-plane.

Use a CAS to solve the problem.

Evaluate the double integral where R is the rectangular region bounded by the lines x = *1, x = 2, y = 0, and y = 4.

The cylindrical parameterization of is

Use a triple integral to find the volume of the solid enclosed by x² = 4y, y + z = 1, and z = 0.

The equation of the tangent plane to x = u, y = v, z = u + v² where u = 2 and v = 2 is

Evaluate , by first sketching R then reversing the order of integration.

Evaluate the double integral $Evaluate the double integral where R is the rectangle bounded by -1 \le x \le 3 and 0 \le y \le 3.$ where R is the rectangle bounded by -1 $\le$ x $\le$ 3 and 0 $\le$ y $\le$ 3.

Use polar coordinates to evaluate $Use polar coordinates to evaluate where R is the region enclosed by and x \ge 0.$ where R is the region enclosed by $Use polar coordinates to evaluate where R is the region enclosed by and x \ge 0.$ and x $\ge$ 0.

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

A lamina with density δ\deltaδ (x, y) = 2xy + 11 is bounded by x = 2, x = 0, y = 0, y = x. Find its center of mass.

Find the volume of the solid formed by the right hemisphere of .

Find the volume of the solid formed by the right hemisphere of .

A uniform beam 1 m in length is supported at its center by a fulcrum. A mass of 20kg is placed at the left end, a mass of 8kg is placed on the beam 10 m from the left end, and a third mass is placed 4 m from the right end. What mass should the third mass be to achieve equilibrium?

The vector normal to the surface given by , , and when and is

Use cylindrical coordinates to evaluate , where R is the solid enclosed by and .

Compute :

Find a parametric representation of the surface in terms of the parameters r and θ\thetaθ , where (r, θ\thetaθ , z) are the cylindrical coordinates of a point on the surface .

Find the volume of the region given by lying above the xy-plane.

Use a CAS to solve the problem.

Evaluate the double integral where R is the rectangular region bounded by the lines x = *1, x = 2, y = 0, and y = 4.

The cylindrical parameterization of is

Use a triple integral to find the volume of the solid enclosed by x2 = 4y, y + z = 1, and z = 0.

The equation of the tangent plane to x = u, y = v, z = u + v2 where u = 2 and v = 2 is

Evaluate , by first sketching R then reversing the order of integration.

Evaluate the double integral where R is the rectangle bounded by -1 ≤\le≤ x ≤\le≤ 3 and 0 ≤\le≤ y ≤\le≤ 3.

Use polar coordinates to evaluate where R is the region enclosed by and x ≥\ge≥ 0.

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

A lamina with density $\delta$ (x, y) = 2xy + 11 is bounded by x = 2, x = 0, y = 0, y = x. Find its center of mass.

Use a triple integral to find the volume of the solid enclosed by x² = 4y, y + z = 1, and z = 0.

The equation of the tangent plane to x = u, y = v, z = u + v² where u = 2 and v = 2 is

Evaluate the double integral $Evaluate the double integral where R is the rectangle bounded by -1 \le x \le 3 and 0 \le y \le 3.$ where R is the rectangle bounded by -1 $\le$ x $\le$ 3 and 0 $\le$ y $\le$ 3.

Use polar coordinates to evaluate $Use polar coordinates to evaluate where R is the region enclosed by and x \ge 0.$ where R is the region enclosed by $Use polar coordinates to evaluate where R is the region enclosed by and x \ge 0.$ and x $\ge$ 0.