Deck 15: Multiple Integrals

Question 1

Find the Jacobian of the transformation. $$x = 6 \alpha \sin \beta , y = 5 \alpha \cos \beta$$

A) $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - 30 \alpha$
B) $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - 20 \alpha \sin \beta \cos \beta$
C) $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = 9 \alpha$
D) $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - \alpha$
E) $\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = 36 \alpha$

Answer 1

$\frac { \partial ( x , y ) } { \partial ( \alpha , \beta ) } = - 30 \alpha$

Question 2

Use cylindrical coordinates to evaluate the triple integral where E is the solid that lies between the sphere and in the first octant.

Answer 2

Question 3

Use spherical coordinates. Evaluate $\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V$ , where $B$ is the ball with center the origin and radius $5$ .

A) $\frac { 312500 } { 7 } \pi$
B) $\frac { 4374 } { 7 }$
C) $\frac { 4374 } { 7 } \pi$
D) $\frac { 559872 } { 7 } \pi$
E) None of these

Answer 3

$\frac { 312500 } { 7 } \pi$

Question 4

Identify the surface with equation

Question 5

Use spherical coordinates to find the moment of inertia of the solid homogeneous hemisphere of radius $5$ and density 1 about a diameter of its base.

A) 205.13
B) $2616.67$
C) 195.22
D) 213.5
E) 198.08

Question 6

Evaluate the integral by making an appropriate change of variables. Round your answer to two decimal places. R is the parallelogram bounded by the lines .

Question 7

Use the transformation $x = \sqrt { 5 } u - \sqrt { \frac { 5 } { 3 } } v , y = \sqrt { 5 } u + \sqrt { \frac { 5 } { 3 } } v$ to evaluate the integral $\iint _ { R } \left( x ^ { 2 } - x y + y ^ { 2 } \right) d A$ , where R is the region bounded by the ellipse $x ^ { 2 } - x y + y ^ { 2 } = 5$ .

A) $\frac { 5 \pi } { \sqrt { 3 } }$
B) $\frac { 25 } { \sqrt { 3 } }$
C) $\frac { 100 \pi } { 3 ^ { 3 / 2 } }$
D) $\frac { 25 \pi } { \sqrt { 3 } }$
E) $\frac { 5 \pi ^ { 2 } } { \sqrt { 3 } }$

Question 8

Identify the surface with equation

Question 9

Find the Jacobian of the transformation.

Question 10

Evaluate $\iiint _ { T } f ( x , y , z ) d V$ where $f ( x , y , z ) = 7 y$ and T is the region bounded by the paraboloid $y = x ^ { 2 } + z ^ { 2 }$ and the plane $y = 1$

A) $\frac { 7 } { 3 }$ $\pi$
B) $\frac { 1 } { 7 }$ $\pi$
C) $\frac { 49 } { 3 }$ $\pi$
D) $7$ $\pi$

Question 11

Use spherical coordinates to find the volume of the solid that lies within the sphere above the xy-plane and below the cone . Round the answer to two decimal places.

Question 12

Use the given transformation to evaluate the integral. $\iint _ { R } x y d A$ , where R is the region in the first quadrant bounded by the lines $y = x , y = 3 x$ and the hyperbolas $y = 2 , x y = 4 ; x = \frac { u } { v } , y = v$ .

A) 9.447
B) 3.296
C) 8.841
D) 4.447
E) 5.088

Question 13

Use spherical coordinate to find the volume above the cone and inside sphere .

Question 14

Find the moment of inertia with respect to a diameter of the base of a solid hemisphere of radius 3 with constant mass density function

Question 15

Use cylindrical coordinates to evaluate $\iiint _ { T } \sqrt { x ^ { 2 } + y ^ { 2 } } d V$ where T is the solid bounded by the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ and the planes $z = 2$ and $z = 5$

A) $2$ $\pi$
B) $14$ $\pi$
C) $21$ $\pi$
D) $3$ $\pi$

Question 16

Use the given transformation to evaluate the integral. $\iint _ { R } ( x + y ) d A$ , where R is the square with vertices (0, 0), (4, 6), (6, $- 4$ ), (10, 2) and $x = 4 u + 6 v , y = 6 u - 4 v$

A) 208
B) 52
C) 343
D) 42
E) 312

Question 17

Use spherical coordinates to evaluate $\iiint _ { B } \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d V$ where B is the ball $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 8$

A) 512 $\pi$
B) 8 $\pi$
C) 64 $\pi$
D) 1024 $\pi$

Question 18

The sketch of the solid is given below. Given $a = 5$ , write the inequalities that describe it.

A) None of these
B) $r ^ { 2 } - 5 \leq z \leq r ^ { 2 }$
C) $r ^ { 2 } \leq z \leq 5$
D) $r ^ { 2 } \leq z \leq 5 + r ^ { 2 }$
E) $$r ^ { 2 } \leq z \leq 5 - r ^ { 2 }$$

Question 19

Identify the surface with equation

Question 20

Find the mass of a solid hemisphere of radius 5 if the mass density at any point on the solid is directly proportional to its distance from the base of the solid.

A) $\frac { 625 } { 4 }$ k $\pi$
B) $25$ k $\pi$
C) $\frac { 25 } { 4 }$ k $\pi$
D) $\frac { 125 } { 4 }$ k $\pi$

Question 21

Evaluate the integral $\iiint _ { B } f ( x , y , z ) d V$ where $f ( x , y , z ) = x y ^ { 2 } + y z ^ { 2 }$ and $B = \{ ( x , y , z ) \mid 0 \leq x \leq 2 , - 5 \leq y \leq 5,0 \leq z \leq 3 \}$ with respect to x, y, and z, in that order.

A) 120
B) 620
C) 180
D) 500

Question 22

The joint density function for a pair of random variables and is given. Find the value of the constant .

Question 23

Express the integral as an iterated integral of the form where E is the solid bounded by the surfaces

Question 24

Find the moment of inertia about the y-axis for a cube of constant density 3 and side length if one vertex is located at the origin and three edges lie along the coordinate axes.

Question 25

Use cylindrical coordinates to evaluate $\int _ { - 2 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - x ^ { 2 } } } \int _ { 0 } ^ { \sqrt { 16 - x ^ { 2 } - y ^ { 2 } } } z d z d y d x$

A) $12$ $\pi$
B) $112$ $\pi$
C) $\frac { 3 } { 2 }$ $\pi$
D) $14$ $\pi$

Question 26

Evaluate the iterated integral

Question 27

The joint density function for random variables and is for and otherwise. Find the value of the constant .
Round the answer to the nearest thousandth.

Question 28

Find the center of mass of a homogeneous solid bounded by the paraboloid and

Question 29

Calculate the iterated integral. $\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 8 y \sin x d z d y d x$

A) 8
B) $\frac { 3 } { 16 }$
C) $0$
D) $\frac { 8 } { 3 }$
E) None of these

Question 30

Find the region E for which the triple integral is a maximum.

Question 31

Use cylindrical or spherical coordinates, whichever seems more appropriate, to evaluate $\iiint _ { E } z d V$ where E lies above the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and below the plane $z = 2$ .

A) $-$ 160.28
B) 175.37
C) $66.99$
D) 176.38
E) 175.93

Question 32

Evaluate the triple integral. Round your answer to one decimal place. lies under the plane and above the region in the -plane bounded by the curves , and .

Question 33

Use a triple integral to find the volume of the solid bounded by $x = y ^ { 2 }$ and the planes $z = 0$ and $x + z = 3$ .

A) $8.3$
B) $183$
C) $2.5$
D) $15.3$
E) $11.3$

Question 34

Find the mass of the solid S bounded by the paraboloid $z = 6 x ^ { 2 } + 6 y ^ { 2 }$ and the plane $z = 5$ if S has constant density 3.

A) 16.25
B) 15.07
C) 24.91
D) 13.92
E) 19.63

Question 35

Use cylindrical coordinates to evaluate the triple integral $\iiint _ { E } y d V$ where E is the solid that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 3$ and $x ^ { 2 } + y ^ { 2 } = 7$ above the xy-plane and below the plane $z = x + 4$ .

A) 8.57
B) 0
C) 3.4
D) 9.19
E) 0.54

Question 36

Evaluate the triple integral. Round your answer to one decimal place.

Question 37

Use cylindrical coordinates to find the volume of the solid that the cylinder cuts out of the sphere of radius 3 centered at the origin.

Question 38

Find the mass of the solid E, if E is the cube given by and the density function is .

Question 39

Express the volume of the wedge in the first octant that is cut from the cylinder by the planes and as an iterated integral with respect to , then to , then to .

Question 40

Use cylindrical coordinates to evaluate $\iiint _ { E } \sqrt { x ^ { 2 } + y ^ { 2 } } d V$ where E is the region that lies inside the cylinder $x ^ { 2 } + y ^ { 2 } = 25$ and between the planes $z = - 6 \text { and } z = 5$ . Round the answer to two decimal places.

A) $2878.33$
B) 2218.41
C) 2931.90
D) 2818.41
E) 2431.90

Question 41

Find the area of the surface. The part of the surface that lies within the cylinder .

Question 42

Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides of length $a = 15$ if the density at any point is proportional to the square of the distance from the vertex opposite the hypotenuse. Assume the vertex opposite the hypotenuse is located at $( 0,0 )$ , and that the sides are along the positive axes.

A) $( 6,6 )$
B) $( 6,15 )$
C) $( 5,6 )$
D) $( 15,15 )$
E) None of these

Question 43

Find the area of the part of the plane that lies in the first octant.

Question 44

Find the area of the surface S where S is the part of the plane that lies above the triangular region with vertices , and

Question 45

Sketch the solid whose volume is given by the integral Evaluate the integral.

Question 46

Express the triple integral as an iterated integral in six different ways using different orders of integration where T is the solid bounded by the planes and

Question 47

Find the area of the surface S where S is the part of the surface that lies inside the cylinder

Question 48

Find the area of the surface. The part of the surface $z = 4 - x ^ { 2 } - y ^ { 2 }$ that lies above the xy-plane.

A) $( \sqrt { 17 } - 17 )$
B) $\frac { 1 } { 6 }$ $\pi ( 17 \sqrt { 17 } - 1 )$
C) $\frac { 1 } { 6 }$ $$( \sqrt { 17 } + 1 )$$
D) $\pi ( 17 \sqrt { 17 } - 1 )$
E) $\frac { 1 } { 6 }$ $( 17 \sqrt { 17 } + 1 )$

Question 49

Find the area of the part of hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 9$ .

A) $( 82 \sqrt { 82 } - 3 \sqrt { 5 } ) \pi$
B) $( 82 \sqrt { 82 } + 5 \sqrt { 5 } ) \pi$
C) $\frac { 2 } { 9 }$ $( 82 \sqrt { 82 } - 5 \sqrt { 5 } )$
D) $\frac { 2 } { 9 }$ $( 82 \sqrt { 82 } - 3 \sqrt { 5 } ) \pi$
E) $\frac { 2 } { 9 }$ $( 82 \sqrt { 82 } - 5 \sqrt { 5 } ) \pi$

Question 50

Find the area of the surface S where S is the part of the sphere that lies to the right of the xz-plane and inside the cylinder

Question 51

Find the area of the part of the plane that lies inside the cylinder .

Question 52

Find the area of the surface. The part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ that lies above the plane $z = 1$ .

A) $24 \pi$
B) $\frac { \pi } { 16 }$
C) $24$
D) $\pi$
E) $24 - \pi$

Question 53

Find the area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 25 z$ that lies inside the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ .

A) $11.5 \pi$
B) $9.5 \pi$
C) $25 \pi$
D) $5 \pi$
E) $15 \pi$

Question 54

Sketch the solid bounded by the graphs of the equations and , and then use a triple integral to find the volume of the solid.

Question 55

Find the area of the surface. Round your answer to three decimal places. $z =$ $\frac { 4 } { 3 }$ $\left( x ^ { 2 / 3 } + y ^ { 2 / 3 } \right) , 0 \leq x \leq 5,0 \leq y \leq 3$

A) $70.0492$
B) $62.3702$
C) $60.0492$
D) $80.3702$
E) $85.3702$

Question 56

Sketch the solid whose volume is given by the iterated integral

Question 57

Find the exact area of the surface. $z = x ^ { 2 } + 2 y , 0 \leq x \leq 1,0 \leq y \leq 2$ .

A) $\frac { 5 \ln ( 3 ) } { 4 }$
B) $\frac { 5 } { 4 }$
C) $3 - \ln ( 5 )$
D) $3 + \frac { 5 } { 4 } \ln ( 5 )$
E) $2 + \frac { 5 } { 3 } \ln ( 4 )$

Question 58

Find the area of the surface S where S is the part of the sphere that lies inside the cylinder

Question 59

Set up, but do not evaluate, the iterated integral giving the mass of the solid T bounded by the cylinder in the first octant and the plane having mass density given by

Question 60

Describe the region whose area is given by the integral.

Question 61

Find the mass and the moments of inertia and and the radii of gyration and for the lamina occupying the region R, where R is the region bounded by the graphs of the equations and and having the mass density

Question 62

Find the mass and the center of mass of the lamina occupying the region R, where R is the region bounded by the graphs of and and having the mass density

Question 63

Use polar coordinates to find the volume of the solid inside the cylinder $x ^ { 2 } + y ^ { 2 } = 16$ and the ellipsoid $6 x ^ { 2 } + 6 y ^ { 2 } + z ^ { 2 } = 64$ .

A) $853.187$
B) $903.187$
C) $1003.187$
D) $753.187$
E) $1103.187$

Question 64

Find the mass of the lamina that occupies the region and has the given density function. Round your answer to two decimal places.

Question 65

Find the mass and the center of mass of the lamina occupying the region R, where R is the region bounded by the graphs of the equations and and having the mass density

Question 66

Use a double integral to find the area of the region R where R is bounded by the circle $r = 6 \sin \theta$

A) $36$ $\pi$
B) $9$ $\pi$
C) $18$ $\pi$
D) $6$ $\pi$

Question 67

A swimming pool is circular with a $60$ -ft diameter. The depth is constant along east-west lines and increases linearly from $3$ ft at the south end to $9$ ft at the north end. Find the volume of water in the pool.

A) $5410 \pi \mathrm { ft } ^ { 3 }$
B) $5500 \pi \mathrm { ft } ^ { 3 }$
C) $5400 \pi \mathrm { ft } ^ { 3 }$
D) $5600 \pi \mathrm { ft } ^ { 3 }$
E) $5700 \pi \mathrm { ft } ^ { 3 }$

Question 68

Find the center of mass of the system comprising masses m_k located at the points P_k in a coordinate plane. Assume that mass is measured in grams and distance is measured in centimeters.
m₁ = 4, m₂ = 3, m₃ = 2
P₁(-3, -3), P₂(0, 3), P₃(-2, -1)

Question 69

Find the center of mass of the lamina that occupies the region D and has the given density function, if D is bounded by the parabola $y = 1 - x ^ { 2 }$ and the x-axis. $\rho ( x , y ) = 4 y$

A) $( 0,0.57 )$
B) $( 4,46.44 )$
C) $( 12,0.57 )$
D) $( 8,4 )$
E) None of these

Question 70

Evaluate the integral by changing to polar coordinates. is the region bounded by the semicircle and the -axis.

Question 71

An electric charge is spread over a rectangular region $R = \{ ( x , y ) \mid 0 \leq x \leq 3,0 \leq y \leq 4 \} .$ Find the total charge on R if the charge density at a point $( x , y )$ in R (measured in coulombs per square meter) is $\sigma ( x , y ) = x ^ { 2 } + 4 y ^ { 3 }$

A) $804$ coulombs
B) $91$ coulombs
C) $300$ coulombs
D) $265$ coulombs

Question 72

Use polar coordinates to find the volume of the solid bounded by the paraboloid $z = 7 - 6 x ^ { 2 } - 6 y ^ { 2 }$ and the plane $z = 1$ .

A) $6 \pi$
B) $13 \pi$
C) $3 \pi$
D) $4.5 \pi$
E) $2 \pi$

Question 73

Find the mass and the center of mass of the lamina occupying the region R, where R is the triangular region with vertices $( 0,0 ) \text {, }$ $( 2,5 )$ and $( 4,0 )$ , and having the mass density $\rho ( x , y ) = x$

A) $m =$ $25$ , $( \bar { x } , \bar { y } ) = \left( \frac { 7 } { 3 } , \frac { 5 } { 3 } \right)$
B) $m = 20$ , $( \bar { x } , \bar { y } ) = \left( \frac { 7 } { 3 } , \frac { 5 } { 3 } \right)$
C) $m =$ $25$ , $( \bar { x } , \bar { y } ) = \left( \frac { 5 } { 3 } , \frac { 7 } { 3 } \right)$
D) $$m = 20$$ , $( \bar { x } , \bar { y } ) = \left( \frac { 5 } { 3 } , \frac { 7 } { 3 } \right)$

Question 74

Evaluate the iterated integral by converting to polar coordinates. Round the answer to two decimal places. $\int _ { - 3 } ^ { 3 } \int _ { 0 } ^ { \sqrt { 9 - y ^ { 2 } } } \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } d x d y$ .

A) $152.68$
B) $5.65$
C) $14.14$
D) $48.6$
E) $381.7$

Question 75

A lamina occupies the part of the disk in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.

Question 76

Use polar coordinates to find the volume of the sphere of radius $3$ . Round to two decimal places.

A) $183.33$
B) $113.1$
C) $173.33$
D) $153.33$
E) $133.1$

Question 77

Find the center of mass of the lamina of the region shown if the density of the circular lamina is four times that of the rectangular lamina.

Question 78

Find the mass of the lamina that occupies the region D and has the given density function, if D is bounded by the parabola $x = y ^ { 2 }$ and the line $y = x - 2$ . $\rho ( x , y ) = 3$

A) $\frac { 3 } { 2 }$
B) $2$
C) 27
D) $\frac { 27 } { 2 }$
E) None of these

Question 79

Find the mass and the moments of inertia and and the radii of gyration and for the lamina occupying the region R, where R is the rectangular region with vertices and , and having uniform density

Question 80

Use polar coordinates to find the volume of the solid under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and above the disk $x ^ { 2 } + y ^ { 2 } \leq 9$ .

A) $40.5 \pi$
B) $27 \pi$
C) $81 \pi$
D) $$324 \pi$$
E) $162 \pi$