Question 1

Find the mass of a square lamina if the lamina has vertices (0, 0), (2, 0), (0, 2), and (2, 2), and a density function $\delta$ (x, y) = 9x2y.

Accepted Answer

The answer of Find the mass of a square lamina...

Question 2

The centroid of the solid given by (x + 11)2 + y2 + (z - 3)2 = 9 is

Accepted Answer

A)  (11, 0, 3) 
B)  (-11, 0, 3) 
C)  (0, 0, 0) 
D)  (11, 0, -3) 
E)  (-11, 0, -3) 
A)  (11, 0, 3) 
B)  (-11, 0, 3) 
C)  (0, 0, 0) 
D)  (11, 0, -3) 
E)  (-11, 0, -3)

Question 3

Use a double integral in polar coordinates to find the volume of the solid enclosed by the sphere x2 + y2 + z2 = 36 and the cylinder x2 + y2 = 1.

Accepted Answer

Question 4

Find the volume under the surface   and over the rectangle
R = { (x, y) : 0 $\le$  x $\le$  1, 0 $\le$  y $\le$  3} .

Accepted Answer

The answer of Find the volume under the surface ...

Question 5

Find   , if

Accepted Answer

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

Question 6

Evaluate

Accepted Answer

The answer of Evaluate  ...

Question 7

Evaluate   if $\delta$(r, $\theta$, z) = 5r.

Accepted Answer

A)  60 $\pi$ 
B)     $\pi$ 
C)  25 
D)     $\pi$ 
E)   $\pi$ 
A)  60 $\pi$ 
B)     $\pi$ 
C)  25 
D)     $\pi$ 
E)   $\pi$

Question 8

Evaluate

Accepted Answer

The answer of Evaluate  ...

Question 9

Use a triple integral to find the volume of the solid in the first octant enclosed by the cylinder z = 4 - y2 and the planes y = x, z = 0, x = 0, and y = 2.

Accepted Answer

Question 10

Find the Jacobian if x = 4u + w, y = 7 + vw, and z = u2v + 3.

Accepted Answer

A)  4u2v - 2uvw 
B)  4u2v + 2uw2 
C)  8(u2v + uw2) 
D)  -8(u2v + uw2) 
E)  (u2v + uw2) 
A)  4u2v - 2uvw 
B)  4u2v + 2uw2 
C)  8(u2v + uw2) 
D)  -8(u2v + uw2) 
E)  (u2v + uw2)

Question 11

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 12

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 z = 7xy + 6.

Accepted Answer

x = r cos @#LAT-DLM&, y = r si

Question 13

Use a double integral in polar coordinates to find the volume enclosed by the sphere x2 + y2 + z2 = 16 and the cylinder (x - 2)2 + y2 = 4.

Accepted Answer

Question 14

Find the volume between   and   below the xy-plane.

Accepted Answer

A)    
B)    
C)  8 $\pi$ 
D)  4 $\pi$ 
E)  2 $\pi$ 
A)    
B)    
C)  8 $\pi$ 
D)  4 $\pi$ 
E)  2 $\pi$

Question 15

Find   , if

Accepted Answer

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

Question 16

The surface expressed parametrically by x = r cos $\theta$0, y = r sin $\theta$,   is

Accepted Answer

A)  a sphere 
B)  an ellipsoid 
C)  a paraboloid 
D)  a cone 
E)  a hemisphere 
A)  a sphere 
B)  an ellipsoid 
C)  a paraboloid 
D)  a cone 
E)  a hemisphere

Question 17

Find the area enclosed by the three-petaled rose r = 24 cos 3$\theta$.

Accepted Answer

A)  1.5   
B)  3   
C)  12   
D)  48   
E)  24 $\pi$ 
A)  1.5   
B)  3   
C)  12   
D)  48   
E)  24 $\pi$

Question 18

Find   , if   and   .

Accepted Answer

A)  2u + 1 
B)    
C)    
D)    
E)    
A)  2u + 1 
B)    
C)    
D)    
E)

Question 19

A lamina with density  $\delta$(x, y) = 4xy is bounded by x = 2, x = 0, y = x, y = 0. Find its mass.

Accepted Answer

A)  4 
B)  8 
C)  1 
D)  16 
E)  32 
A)  4 
B)  8 
C)  1 
D)  16 
E)  32

Question 20

Evaluate   , if $\delta$(r, $\theta$, z) = 2z2.

Accepted Answer

A)     $\pi$ 
B)  27   
C)  6 
D)     $\pi$ 
E)     $\pi$ 
A)     $\pi$ 
B)  27   
C)  6 
D)     $\pi$ 
E)     $\pi$

Find the mass of a square lamina if the lamina has vertices (0, 0), (2, 0), (0, 2), and (2, 2), and a density function $\delta$ (x, y) = 9x²y.

The centroid of the solid given by (x + 11)² + y² + (z - 3)² = 9 is

Use a double integral in polar coordinates to find the volume of the solid enclosed by the sphere x² + y² + z² = 36 and the cylinder x² + y² = 1.

Find the volume under the surface $Find the volume under the surface and over the rectangle R = { (x, y) : 0 \le x \le 1, 0 \le y \le 3} .$ and over the rectangle R = { (x, y) : 0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 3} .

Find , if

Evaluate

Evaluate $Evaluate if \delta (r, \theta , z) = 5r.$ if $\delta$ (r, $\theta$ , z) = 5r.

Evaluate

Use a triple integral to find the volume of the solid in the first octant enclosed by the cylinder z = 4 - y² and the planes y = x, z = 0, x = 0, and y = 2.

Find the Jacobian if x = 4u + w, y = 7 + vw, and z = u²v + 3.

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 z = 7xy + 6.

Use a double integral in polar coordinates to find the volume enclosed by the sphere x² + y² + z² = 16 and the cylinder (x - 2)² + y² = 4.

Find the volume between and below the xy-plane.

Find , if

The surface expressed parametrically by x = r cos $\theta$ 0, y = r sin $\theta$ , $The surface expressed parametrically by x = r cos \theta 0, y = r sin \theta , is$ is

Find the area enclosed by the three-petaled rose r = 24 cos 3 $\theta$ .

Find , if and .

A lamina with density $\delta$ (x, y) = 4xy is bounded by x = 2, x = 0, y = x, y = 0. Find its mass.

Evaluate $Evaluate , if \delta (r, \theta , z) = 2z<sup>2</sup>.$ , if $\delta$ (r, $\theta$ , z) = 2z².

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

Find the mass of a square lamina if the lamina has vertices (0, 0), (2, 0), (0, 2), and (2, 2), and a density function δ\deltaδ (x, y) = 9x2y.

The centroid of the solid given by (x + 11)2 + y2 + (z - 3)2 = 9 is

Use a double integral in polar coordinates to find the volume of the solid enclosed by the sphere x2 + y2 + z2 = 36 and the cylinder x2 + y2 = 1.

Find the volume under the surface and over the rectangle R = { (x, y) : 0 ≤\le≤ x ≤\le≤ 1, 0 ≤\le≤ y ≤\le≤ 3} .

Find , if

Evaluate

Evaluate if δ\deltaδ (r, θ\thetaθ , z) = 5r.

Evaluate

Use a triple integral to find the volume of the solid in the first octant enclosed by the cylinder z = 4 - y2 and the planes y = x, z = 0, x = 0, and y = 2.

Find the Jacobian if x = 4u + w, y = 7 + vw, and z = u2v + 3.

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 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 z = 7xy + 6.

Use a double integral in polar coordinates to find the volume enclosed by the sphere x2 + y2 + z2 = 16 and the cylinder (x - 2)2 + y2 = 4.

Find the volume between and below the xy-plane.

Find , if

The surface expressed parametrically by x = r cos θ\thetaθ 0, y = r sin θ\thetaθ , is

Find the area enclosed by the three-petaled rose r = 24 cos 3 θ\thetaθ .

Find , if and .

A lamina with density δ\deltaδ (x, y) = 4xy is bounded by x = 2, x = 0, y = x, y = 0. Find its mass.

Evaluate , if δ\deltaδ (r, θ\thetaθ , z) = 2z2.

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

Find the mass of a square lamina if the lamina has vertices (0, 0), (2, 0), (0, 2), and (2, 2), and a density function $\delta$ (x, y) = 9x²y.

The centroid of the solid given by (x + 11)² + y² + (z - 3)² = 9 is

Use a double integral in polar coordinates to find the volume of the solid enclosed by the sphere x² + y² + z² = 36 and the cylinder x² + y² = 1.

Find the volume under the surface $Find the volume under the surface and over the rectangle R = { (x, y) : 0 \le x \le 1, 0 \le y \le 3} .$ and over the rectangle R = { (x, y) : 0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 3} .

Evaluate $Evaluate if \delta (r, \theta , z) = 5r.$ if $\delta$ (r, $\theta$ , z) = 5r.

Use a triple integral to find the volume of the solid in the first octant enclosed by the cylinder z = 4 - y² and the planes y = x, z = 0, x = 0, and y = 2.

Find the Jacobian if x = 4u + w, y = 7 + vw, and z = u²v + 3.

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 z = 7xy + 6.

Use a double integral in polar coordinates to find the volume enclosed by the sphere x² + y² + z² = 16 and the cylinder (x - 2)² + y² = 4.

The surface expressed parametrically by x = r cos $\theta$ 0, y = r sin $\theta$ , $The surface expressed parametrically by x = r cos \theta 0, y = r sin \theta , is$ is

Find the area enclosed by the three-petaled rose r = 24 cos 3 $\theta$ .

A lamina with density $\delta$ (x, y) = 4xy is bounded by x = 2, x = 0, y = x, y = 0. Find its mass.

Evaluate $Evaluate , if \delta (r, \theta , z) = 2z<sup>2</sup>.$ , if $\delta$ (r, $\theta$ , z) = 2z².