Question 1

Set up the volume integral for the solid cylinder $x ^ { 2 } + y ^ { 2 } \leq 1$ , bounded above by $z = x ^ { 3 } + y ^ { 2 } = 9$ and below by z = 2x + 3y + 3.

Accepted Answer

The answer of Set up the volume integral for the...

Question 2

Evaluate the iterated integral $\int _ { 0 } ^ { x / 4 } \int _ { 0 } ^ { 2 } \int _ { 0 } ^ { x ^ { 2 } } x \cos ( 2 y ) d z d x d y$ .

Accepted Answer

A) 0 
B) 8 
C) 4 
D) 16 
E) 6
F)2
G)5
H)1 
A) 0 
B) 8 
C) 4 
D) 16 
E) 6
F)2
G)5
H)1

Question 3

Find the Jacobian of the transformation x = 2u, $y = 3 v ^ { 2 }$ , $z = 4 w ^ { 3 }$ .

Accepted Answer

The answer of Find the Jacobian of the transformation x...

Question 4

Find the surface area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ that lies above the plane z = 1.

Accepted Answer

A)  $( 4 - 2 \sqrt { 2 } ) \pi$ 
B) 2 
C) 2$\pi$ 
D) 4$\pi$ 
E)  $( 8 - 4 \sqrt { 3 } ) \pi$
F)6$\pi$
G)8$\pi$
H)16$\pi$ 
A)  $( 4 - 2 \sqrt { 2 } ) \pi$ 
B) 2 
C) 2$\pi$ 
D) 4$\pi$ 
E)  $( 8 - 4 \sqrt { 3 } ) \pi$
F)6$\pi$
G)8$\pi$
H)16$\pi$

Question 5

Compute the average value of $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ over $R = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq 2 \}$ .

Accepted Answer

The answer of Compute the average value of \(f (...

Question 6

Evaluate the iterated integral $\int _ { 0 } ^ { x / 2 } \int _ { 0 } ^ { x / 2 } \int _ { 0 } ^ { 1 } \rho ^ { 3 } \sin \phi \cos \phi d \rho d \phi d \theta$ .

Accepted Answer

A)  $\frac { \pi } { 8 }$ 
B)  $\frac { \pi } { 4 }$ 
C) $\pi$ 
D) 2$\pi$ 
E)  $\frac { \pi } { 16 }$

F) $\frac { \pi } { 2 }$

G) $\frac { \pi } { 3 }$

H) $\frac { \pi } { 6 }$ 
A)  $\frac { \pi } { 8 }$ 
B)  $\frac { \pi } { 4 }$ 
C) $\pi$ 
D) 2$\pi$ 
E)  $\frac { \pi } { 16 }$

F) $\frac { \pi } { 2 }$

G) $\frac { \pi } { 3 }$

H) $\frac { \pi } { 6 }$

Question 7

Evaluate the iterated integral $\int _ { 0 } ^ { \mathrm { In  } 4 } \int _ { 0 } ^ { \mathrm { In } 3 } \int _ { 0 } ^ { \mathrm { In  } 2 } e ^ { 05 x + y - z } d z d y d x$ .

Accepted Answer

A) 6 
B) 1 
C) 3 
D) 2 
E) -6
F)-1
G)-3
H)-2 
A) 6 
B) 1 
C) 3 
D) 2 
E) -6
F)-1
G)-3
H)-2

Question 8

Convert the integral $\int _ { 0 } ^ { 2 } \int _ { - \sqrt { 4 - x ^ { 2 } } } ^ { 0 } \frac { x y } { \sqrt { x ^ { 2 } + y ^ { 2 } } } d y d x$ to polar coordinates and evaluate it.

Accepted Answer

The answer of Convert the integral \(\int _ { 0...

Question 9

The centroid (x, y) of a planar region D is the center of mass with the density function $p ( x , y ) = 1$ . Find the centroid of $D = \left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 4 , x \geq 0 , y \geq 0 \right\}$ .

Accepted Answer

The answer of The centroid (x, y) of a planar...

Question 10

Find the mass of the solid that occupies the region bounded by the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and the plane z = 1 and has density function $\rho ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Accepted Answer

A)  $\frac { \pi } { 4 }$ 
B)  $\frac { \pi } { 3 }$ 
C)  $\frac { \pi } { 2 }$ 
D)  $\frac { 2 \pi } { 3 }$ 
E)  $\frac { 3 \pi } { 4 }$
  
F)$\pi$

G) $\frac { 4 \pi } { 15 }$

H)2$\pi$ 
A)  $\frac { \pi } { 4 }$ 
B)  $\frac { \pi } { 3 }$ 
C)  $\frac { \pi } { 2 }$ 
D)  $\frac { 2 \pi } { 3 }$ 
E)  $\frac { 3 \pi } { 4 }$
  
F)$\pi$

G) $\frac { 4 \pi } { 15 }$

H)2$\pi$

Question 11

Let E be the solid bounded above by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ and below by the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .(a) Express the volume of E as an iterated integral in rectangular coordinates.(b) Express the volume of E as an iterated integral in cylindrical coordinates.(c) Express the volume of E as an iterated integral in spherical coordinates.

Accepted Answer

The answer of Let E be the solid bounded above...

Question 12

Find the area of the part of the hemisphere $z = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } }$ that lies above the area inside the circle $x ^ { 2 } + y ^ { 2 } = 1$ .

Accepted Answer

A)  $( 4 - 2 \sqrt { 2 } ) \pi$ 
B)  $( 6 - 3 \sqrt { 2 } ) \pi$ 
C)  $( 8 - 4 \sqrt { 2 } ) \pi$ 
D)  $( 6 - 3 \sqrt { 3 } ) \pi$ 
E)  $( 8 - 4 \sqrt { 3 } ) \pi$

F)6$\pi$
 
G)8$\pi$

H)16$\pi$ 
A)  $( 4 - 2 \sqrt { 2 } ) \pi$ 
B)  $( 6 - 3 \sqrt { 2 } ) \pi$ 
C)  $( 8 - 4 \sqrt { 2 } ) \pi$ 
D)  $( 6 - 3 \sqrt { 3 } ) \pi$ 
E)  $( 8 - 4 \sqrt { 3 } ) \pi$

F)6$\pi$
 
G)8$\pi$

H)16$\pi$

Question 13

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { x } ^ { 3 x } \int _ { \sqrt { y } } ^ { x } 2 z d z d y d x$ .

Accepted Answer

A)  $- \frac { 5 } { 12 }$ 
B)  $- \frac { 5 } { 6 }$ 
C)  $- \frac { 1 } { 12 }$ 
D)  $- \frac { 1 } { 6 }$ 
E) 
$\frac { 5 } { 12 }$ 
F) $\frac { 5 } { 6 }$

G) $\frac { 1 } { 12 }$

H) $\frac { 1 } { 6 }$ 
A)  $- \frac { 5 } { 12 }$ 
B)  $- \frac { 5 } { 6 }$ 
C)  $- \frac { 1 } { 12 }$ 
D)  $- \frac { 1 } { 6 }$ 
E) 
$\frac { 5 } { 12 }$ 
F) $\frac { 5 } { 6 }$

G) $\frac { 1 } { 12 }$

H) $\frac { 1 } { 6 }$

Question 14

Find the average value of the function f(x, y, z) = xyz over the solid E bounded by planes z = y, y = x, x = 1, and z = 0.

Accepted Answer

The answer of Find the average value of the function...

Question 15

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

Accepted Answer

The answer of Find the area of that part of...

Question 16

Find the Jacobian of the transformation x = u + v, y = 2u - v.

Accepted Answer

A) 1 
B) 2 
C) 3 
D) 4 
E) -1
F)-2
G)-3
H)-4 
A) 1 
B) 2 
C) 3 
D) 4 
E) -1
F)-2
G)-3
H)-4

Question 17

Find the moments of inertia $I _ { x }$ , $I _ { y }$ , and $I _ { 0 }$ for the lamina that occupies the region given by $D = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq 3 \}$ .

Accepted Answer

The answer of Find the moments of inertia \(I _...

Question 18

Evaluate the triple integral $\iiint _ { E } ( x + 2 y ) d V$ , where $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ .

Accepted Answer

A)  $\frac { 1 } { 4 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 3 } { 4 }$ 
D) 1 
E)  $\frac { 5 } { 4 }$ 
F) $\frac { 3 } { 2 }$ 
 
G) $\frac { 7 } { 4 }$ 
H)2 
A)  $\frac { 1 } { 4 }$ 
B)  $\frac { 1 } { 2 }$ 
C)  $\frac { 3 } { 4 }$ 
D) 1 
E)  $\frac { 5 } { 4 }$ 
F) $\frac { 3 } { 2 }$ 
 
G) $\frac { 7 } { 4 }$ 
H)2

Question 19

Evaluate the double integral $\int _ { 0 } ^ { 1 } \int _ { x } ^ { 1 } \sin y ^ { 2 } d y d x$ .

Accepted Answer

The answer of Evaluate the double integral \(\int _ {...

Question 20

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { \sqrt { x } } ^ { 1 } \cos \left( \pi y ^ { 3 } \right) d y d x$ .

Accepted Answer

A)  $( \cos 1 - 1 )$ 
B) 
$\frac { 1 } { 3 } ( 1 - \cos 1 )$ 
C)  $\frac { 1 } { 3 } \cos 1$ 
D) 1 
E)  $( \sin 1 - 1 )$ 
F)
$\frac { 1 } { 3 } ( 1 - \sin 1 )$

G) $\frac { 1 } { 3 } \sin 1$ 
H)0 
A)  $( \cos 1 - 1 )$ 
B) 
$\frac { 1 } { 3 } ( 1 - \cos 1 )$ 
C)  $\frac { 1 } { 3 } \cos 1$ 
D) 1 
E)  $( \sin 1 - 1 )$ 
F)
$\frac { 1 } { 3 } ( 1 - \sin 1 )$

G) $\frac { 1 } { 3 } \sin 1$ 
H)0

Set up the volume integral for the solid cylinder $x ^ { 2 } + y ^ { 2 } \leq 1$ , bounded above by $z = x ^ { 3 } + y ^ { 2 } = 9$ and below by z = 2x + 3y + 3.

Evaluate the iterated integral $\int _ { 0 } ^ { x / 4 } \int _ { 0 } ^ { 2 } \int _ { 0 } ^ { x ^ { 2 } } x \cos ( 2 y ) d z d x d y$ .

Find the Jacobian of the transformation x = 2u, $y = 3 v ^ { 2 }$ , $z = 4 w ^ { 3 }$ .

Find the surface area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ that lies above the plane z = 1.

Compute the average value of $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ over $R = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq 2 \}$ .

Evaluate the iterated integral $\int _ { 0 } ^ { x / 2 } \int _ { 0 } ^ { x / 2 } \int _ { 0 } ^ { 1 } \rho ^ { 3 } \sin \phi \cos \phi d \rho d \phi d \theta$ .

Evaluate the iterated integral $\int _ { 0 } ^ { \mathrm { In } 4 } \int _ { 0 } ^ { \mathrm { In } 3 } \int _ { 0 } ^ { \mathrm { In } 2 } e ^ { 05 x + y - z } d z d y d x$ .

Convert the integral $\int _ { 0 } ^ { 2 } \int _ { - \sqrt { 4 - x ^ { 2 } } } ^ { 0 } \frac { x y } { \sqrt { x ^ { 2 } + y ^ { 2 } } } d y d x$ to polar coordinates and evaluate it.

The centroid (x, y) of a planar region D is the center of mass with the density function $p ( x , y ) = 1$ . Find the centroid of $D = \left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 4 , x \geq 0 , y \geq 0 \right\}$ .

Find the mass of the solid that occupies the region bounded by the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and the plane z = 1 and has density function $\rho ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Find the area of the part of the hemisphere $z = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } }$ that lies above the area inside the circle $x ^ { 2 } + y ^ { 2 } = 1$ .

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { x } ^ { 3 x } \int _ { \sqrt { y } } ^ { x } 2 z d z d y d x$ .

Find the average value of the function f(x, y, z) = xyz over the solid E bounded by planes z = y, y = x, x = 1, and z = 0.

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

Find the Jacobian of the transformation x = u + v, y = 2u - v.

Find the moments of inertia $I _ { x }$ , $I _ { y }$ , and $I _ { 0 }$ for the lamina that occupies the region given by $D = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq 3 \}$ .

Evaluate the triple integral $\iiint _ { E } ( x + 2 y ) d V$ , where $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ .

Evaluate the double integral $\int _ { 0 } ^ { 1 } \int _ { x } ^ { 1 } \sin y ^ { 2 } d y d x$ .

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { \sqrt { x } } ^ { 1 } \cos \left( \pi y ^ { 3 } \right) d y d x$ .

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Exam 12: Multiple Integrals

Set up the volume integral for the solid cylinder x2+y2≤1x ^ { 2 } + y ^ { 2 } \leq 1x2+y2≤1 , bounded above by z=x3+y2=9z = x ^ { 3 } + y ^ { 2 } = 9z=x3+y2=9 and below by z = 2x + 3y + 3.

Evaluate the iterated integral ∫0x/4∫02∫0x2xcos⁡(2y)dzdxdy\int _ { 0 } ^ { x / 4 } \int _ { 0 } ^ { 2 } \int _ { 0 } ^ { x ^ { 2 } } x \cos ( 2 y ) d z d x d y∫0x/4​∫02​∫0x2​xcos(2y)dzdxdy .

Find the Jacobian of the transformation x = 2u, y=3v2y = 3 v ^ { 2 }y=3v2 , z=4w3z = 4 w ^ { 3 }z=4w3 .

Find the surface area of the part of the sphere x2+y2+z2=4x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4x2+y2+z2=4 that lies above the plane z = 1.

Compute the average value of f(x,y)=x2+y2f ( x , y ) = x ^ { 2 } + y ^ { 2 }f(x,y)=x2+y2 over R={(x,y)∣0≤x≤2,0≤y≤2}R = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq 2 \}R={(x,y)∣0≤x≤2,0≤y≤2} .

Evaluate the iterated integral ∫0x/2∫0x/2∫01ρ3sin⁡ϕcos⁡ϕdρdϕdθ\int _ { 0 } ^ { x / 2 } \int _ { 0 } ^ { x / 2 } \int _ { 0 } ^ { 1 } \rho ^ { 3 } \sin \phi \cos \phi d \rho d \phi d \theta∫0x/2​∫0x/2​∫01​ρ3sinϕcosϕdρdϕdθ .

Evaluate the iterated integral ∫0In4∫0In3∫0In2e05x+y−zdzdydx\int _ { 0 } ^ { \mathrm { In } 4 } \int _ { 0 } ^ { \mathrm { In } 3 } \int _ { 0 } ^ { \mathrm { In } 2 } e ^ { 05 x + y - z } d z d y d x∫0In4​∫0In3​∫0In2​e05x+y−zdzdydx .

Convert the integral ∫02∫−4−x20xyx2+y2dydx\int _ { 0 } ^ { 2 } \int _ { - \sqrt { 4 - x ^ { 2 } } } ^ { 0 } \frac { x y } { \sqrt { x ^ { 2 } + y ^ { 2 } } } d y d x∫02​∫−4−x2​0​x2+y2​xy​dydx to polar coordinates and evaluate it.

The centroid (x, y) of a planar region D is the center of mass with the density function p(x,y)=1p ( x , y ) = 1p(x,y)=1 . Find the centroid of D={(x,y)∣x2+y2≤4,x≥0,y≥0}D = \left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 4 , x \geq 0 , y \geq 0 \right\}D={(x,y)∣x2+y2≤4,x≥0,y≥0} .

Find the mass of the solid that occupies the region bounded by the paraboloid z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 and the plane z = 1 and has density function ρ(x,y,z)=x2+y2\rho ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } }ρ(x,y,z)=x2+y2​ .

Find the area of the part of the hemisphere z=4−x2−y2z = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } }z=4−x2−y2​ that lies above the area inside the circle x2+y2=1x ^ { 2 } + y ^ { 2 } = 1x2+y2=1 .

Evaluate the iterated integral ∫01∫x3x∫yx2zdzdydx\int _ { 0 } ^ { 1 } \int _ { x } ^ { 3 x } \int _ { \sqrt { y } } ^ { x } 2 z d z d y d x∫01​∫x3x​∫y​x​2zdzdydx .

Find the average value of the function f(x, y, z) = xyz over the solid E bounded by planes z = y, y = x, x = 1, and z = 0.

Find the area of that part of the paraboloid z=4−x2−y2z = 4 - x ^ { 2 } - y ^ { 2 }z=4−x2−y2 that lies above the xy-plane.

Find the Jacobian of the transformation x = u + v, y = 2u - v.

Find the moments of inertia IxI _ { x }Ix​ , IyI _ { y }Iy​ , and I0I _ { 0 }I0​ for the lamina that occupies the region given by D={(x,y)∣0≤x≤2,0≤y≤3}D = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq 3 \}D={(x,y)∣0≤x≤2,0≤y≤3} .

Evaluate the triple integral ∭E(x+2y)dV\iiint _ { E } ( x + 2 y ) d V∭E​(x+2y)dV , where E={(x,y,z)∣0≤x≤1,0≤y≤1,0≤z≤1}E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}E={(x,y,z)∣0≤x≤1,0≤y≤1,0≤z≤1} .

Evaluate the double integral ∫01∫x1sin⁡y2dydx\int _ { 0 } ^ { 1 } \int _ { x } ^ { 1 } \sin y ^ { 2 } d y d x∫01​∫x1​siny2dydx .

Evaluate the iterated integral ∫01∫x1cos⁡(πy3)dydx\int _ { 0 } ^ { 1 } \int _ { \sqrt { x } } ^ { 1 } \cos \left( \pi y ^ { 3 } \right) d y d x∫01​∫x​1​cos(πy3)dydx .

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Set up the volume integral for the solid cylinder $x ^ { 2 } + y ^ { 2 } \leq 1$ , bounded above by $z = x ^ { 3 } + y ^ { 2 } = 9$ and below by z = 2x + 3y + 3.

Evaluate the iterated integral $\int _ { 0 } ^ { x / 4 } \int _ { 0 } ^ { 2 } \int _ { 0 } ^ { x ^ { 2 } } x \cos ( 2 y ) d z d x d y$ .

Find the Jacobian of the transformation x = 2u, $y = 3 v ^ { 2 }$ , $z = 4 w ^ { 3 }$ .

Find the surface area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ that lies above the plane z = 1.

Compute the average value of $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ over $R = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq 2 \}$ .

Evaluate the iterated integral $\int _ { 0 } ^ { x / 2 } \int _ { 0 } ^ { x / 2 } \int _ { 0 } ^ { 1 } \rho ^ { 3 } \sin \phi \cos \phi d \rho d \phi d \theta$ .

Evaluate the iterated integral $\int _ { 0 } ^ { \mathrm { In } 4 } \int _ { 0 } ^ { \mathrm { In } 3 } \int _ { 0 } ^ { \mathrm { In } 2 } e ^ { 05 x + y - z } d z d y d x$ .

Convert the integral $\int _ { 0 } ^ { 2 } \int _ { - \sqrt { 4 - x ^ { 2 } } } ^ { 0 } \frac { x y } { \sqrt { x ^ { 2 } + y ^ { 2 } } } d y d x$ to polar coordinates and evaluate it.

The centroid (x, y) of a planar region D is the center of mass with the density function $p ( x , y ) = 1$ . Find the centroid of $D = \left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 4 , x \geq 0 , y \geq 0 \right\}$ .

Find the mass of the solid that occupies the region bounded by the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and the plane z = 1 and has density function $\rho ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Find the area of the part of the hemisphere $z = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } }$ that lies above the area inside the circle $x ^ { 2 } + y ^ { 2 } = 1$ .

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { x } ^ { 3 x } \int _ { \sqrt { y } } ^ { x } 2 z d z d y d x$ .

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

Find the moments of inertia $I _ { x }$ , $I _ { y }$ , and $I _ { 0 }$ for the lamina that occupies the region given by $D = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq 3 \}$ .

Evaluate the triple integral $\iiint _ { E } ( x + 2 y ) d V$ , where $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ .

Evaluate the double integral $\int _ { 0 } ^ { 1 } \int _ { x } ^ { 1 } \sin y ^ { 2 } d y d x$ .

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { \sqrt { x } } ^ { 1 } \cos \left( \pi y ^ { 3 } \right) d y d x$ .