Deck 14: Multiple Integrals

Question 1

Find , if and

A)
B)
C)
D)
E)

Answer 1

Question 2

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

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

Answer 2

8

Question 3

Find , if and .

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

Answer 3

Question 4

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

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

Question 5

Find the centroid of the lamina enclosed by x = 5(4y - y²)and the y-axis.

Question 6

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.

Question 7

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 x-axis.

A)
B) 8
C)
D)
E)

Question 8

Find the centroid of the lamina enclosed by y = 2(4 - x), x = 0, and y = 0.

Question 9

Find the center of gravity of the lamina enclosed by x = 0, x = 4, y = 0, and y = 3 if its density is given by $\delta$ (x, y) = 5(x + y²).

Question 10

Find the centroid of the lamina enclosed by y = x² and the line y = 4.

Question 11

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.

A)
B)
C)
D)
E)

Question 12

Find the Jacobian if x = 5u + w, y = vw, and z = u²v - 9.

A) 5u²v - 2uw
B) 5u²v + 2uw²
C) 10(u²v + uw²)
D) -10(u²v + uw²)
E) 5u²v - 2uvw

Question 13

The centroid of a rectangular solid in the first octant with vertices (0, 0, 0), (0, 11, 0), and (11, 0, 11) is

A)
B) (1, 1, 1)
C)
D)
E)

Question 14

The centroid of a rectangular solid in the first octant with vertices (0, 0, 0), (0, 0, 4), and (4, 4, 4) is

A)
B) (0, 2, 4)
C) (2, 2, 2)
D)
E)

Question 15

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.

Question 16

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?

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

Question 17

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.

Question 18

Find the Jacobian if and .

A) 0
B)
C)
D)
E)

Question 19

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.

A)
B)
C)
D)
E)

Question 20

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.

A)
B) 16
C)
D)
E)

Question 21

Find , if

A)
B)
C)
D)
E)

Question 22

Find the Jacobian, ; x = uv³, y = 8 + uv.

Question 23

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

A)
B)
C)
D)
E)

Question 24

Find the Jacobian if , and .

A)
B)
C)
D)
E)

Question 25

Find , if

A)
B)
C)
D)
E)

Question 26

Find the Jacobian, ; x = 3uv + w, y = u + 2v + 3w, z = u - v + 6w + 11.

Question 27

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

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

Question 28

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

Question 29

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

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

Question 30

Find the surface area of the portion of the cone that is above the region in the first quadrant bounded by the line , and the parabola .

A)
B)
C)
D)
E)

Question 31

Find the Jacobian, ; x = 5 + e ^uv , y = uv².

Question 32

Find the Jacobian, ; u = 8 + 3x + y, v = 2x - y.

Question 33

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

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

Question 34

Find the Jacobian, ; x = 3u + 2, y = uv.

Question 35

Find , if

A)
B)
C)
D)
E)

Question 36

Find the Jacobian, ; x = 7e ^uv , y = uv⁶.

Question 37

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

Question 38

Use an appropriate transform to find the area of the region in the first quadrant enclosed by x + y = 1, x + y = 2, 3x - 2y = 2, and 3x - 2y = 5.

Question 39

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

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

Question 40

Use spherical coordinates to find the mass of the solid bounded below by and above by if its density is given by .

A)
B)
C)
D)
E)

Question 41

The cylindrical parameterization of is

A) x = r cos $\theta$ , y = r sin $\theta$ ,
B) x = r sin $\theta$ , y = r cos $\theta$ ,
C) x = r cos $\theta$ , y = r sin $\theta$ ,
D) x = r sin $\theta$ , y = r cos $\theta$ , z = 2
E) x = r cos $\theta$ , y = r sin $\theta$ ,

Question 42

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

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

Question 43

Use cylindrical coordinates to find the mass of the solid bounded below by and above by if its density is given by .

A)

B)

C)

D)

E)

Question 44

Evaluate

Question 45

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

Question 46

The surface expressed parametrically by x = r cos $\theta$ , y = r sin $\theta$ , z = 90 - r² is

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

Question 47

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

A)
B)
C)
D)
E)

Question 48

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 and x = r cos $\theta$ , y = r sin $\theta$ on the surface is

A) z = r cos $\theta$ e ^r
B)
C) z = cos $\theta$ e ^r
D)
E)

Question 49

Use a triple integral to find the volume of the tetrahedron enclosed by 10x + 10y + z = 2 and the coordinate planes.

Question 50

The cylindrical parameterization of is

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 $\theta$ , z = 3e ^{r sin $\theta$}
D) x = r sin $\theta$ , y = r cos $\theta$ , z = 3e ^{r sin $\theta$}
E) x = r cos $\theta$ , y = r sin $\theta$ , z = 3r²e ^{r sin $\theta$}

Question 51

The surface expressed parametrically by , , and is:

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

Question 52

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

A)
B)
C)
D)
E)

Question 53

The cylindrical parameterization of is

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 54

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

Question 55

Use a triple integral to find the volume of the solid in the first octant enclosed by
z = x² + y², y = x, and x = 1.

Question 56

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.

Question 57

Use a triple integral to find the volume of the solid in the first octant enclosed by z = y, y² = x, and x = 1.

Question 58

Compute :

Question 59

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.

Question 60

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

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 61

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

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

Question 62

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 .

Question 63

Use a double integral in polar coordinates to find the volume in the first octant of the solid enclosed by x² + y² = 16, y = z, and z = 0.

Question 64

Evaluate . Hint: First convert to cylindrical coordinates.

A)
B)
C)
D)
E)

Question 65

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

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

Question 66

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.

Question 67

Find the volume between and below the xy-plane.

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

Question 68

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

Question 69

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.

Question 70

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 .

Question 71

Evaluate .

A)
B)
C)
D)
E)

Question 72

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 = 12xy.

Question 73

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

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

Question 74

Find the surface area of the portion of the cone that is above the region in the first quadrant bounded by the line , and the parabola .

A)
B)
C)
D)
E)

Question 75

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 .

Question 76

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

A)
B)
C)
D)
E)

Question 77

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

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

Question 78

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

A)
B)
C)
D)
E)

Question 79

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 .

Question 80

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

A)
B)
C)
D)
E)