Question 1

Evaluate the double integral $\iint _ { R } \frac { y } { ( x + 1 ) ^ { 2 } } d A$ , where $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ .

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 2 } { 3 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  $\frac { 3 } { 4 }$ 
E) $\frac { 2 } { 3 }$.

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

G) $\frac { 5 } { 6 }$

H) $\frac { 4 } { 3 }$ 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 2 } { 3 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  $\frac { 3 } { 4 }$ 
E) $\frac { 2 } { 3 }$.

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

G) $\frac { 5 } { 6 }$

H) $\frac { 4 } { 3 }$

Question 2

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

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 1 } { 3 }$ 
C) 
$\frac { 1 } { 4 }$ 
D)  $\frac { 3 } { 4 }$ 
E)  $\frac { 2 } { 3 }$   
F) $\frac { 1 } { 6 }$

G) $\frac { 5 } { 6 }$

H)1 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 1 } { 3 }$ 
C) 
$\frac { 1 } { 4 }$ 
D)  $\frac { 3 } { 4 }$ 
E)  $\frac { 2 } { 3 }$   
F) $\frac { 1 } { 6 }$

G) $\frac { 5 } { 6 }$

H)1

Question 3

Find the mass of the solid that occupies the region $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ and has density function $\rho ( x , y , z ) = x$ .

Accepted Answer

A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $\frac { 2 } { 3 }$ 
F) $\frac { 3 } { 4 }$

G) $\frac { 5 } { 6 }$

H)1 
A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $\frac { 2 } { 3 }$ 
F) $\frac { 3 } { 4 }$

G) $\frac { 5 } { 6 }$

H)1

Question 4

Let $f ( x , y ) = x y$ , and let $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ . Let R be partitioned into four subrectangles by the lines $x = \frac { 1 } { 2 }$ and $y = \frac { 1 } { 2 }$ , and let $\left( x _ { i } ^ { * } , y _ { j } ^ { * } \right)$ be the upper left corner of Rij. Calculate the double Riemann sum of f.

Accepted Answer

A)  $\frac { 3 } { 16 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 5 } { 16 }$ 
D)  $\frac { 3 } { 8 }$ 
E) $\frac { 7 } { 16 }$ 
F)$\frac { 1 } { 2 }$

G) $\frac { 9 } { 16 }$

H) $\frac { 3 } { 4 }$ 
A)  $\frac { 3 } { 16 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 5 } { 16 }$ 
D)  $\frac { 3 } { 8 }$ 
E) $\frac { 7 } { 16 }$ 
F)$\frac { 1 } { 2 }$

G) $\frac { 9 } { 16 }$

H) $\frac { 3 } { 4 }$

Question 5

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

Accepted Answer

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

Question 6

Find the Jacobian of the transformation $x = r \cos \theta$ , $y = r \sin \theta$ , z = z.

Accepted Answer

A) $\theta$ 
B) z 
C) r 
D) 1 
E) 2$\theta$
F)2z
G)2r
H)2 
A) $\theta$ 
B) z 
C) r 
D) 1 
E) 2$\theta$
F)2z
G)2r
H)2

Question 7

Use a triple integral in spherical coordinates to find the volume of that part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ which lies inside the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Accepted Answer

The answer of Use a triple integral in spherical coordinates...

Question 8

Find the area of the portion of the surface $z = x ^ { 2 } + y ^ { 2 }$ inside the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ .

Accepted Answer

The answer of Find the area of the portion of...

Question 9

Evaluate $\iint _ { R } \frac { 1 } { x } d A$ , where R is the region bounded by the curves $x ^ { 2 } + y ^ { 2 } = 1$ , $y = x$ , $x = 2$ , and $y = 0$ .

Accepted Answer

The answer of Evaluate \(\iint _ { R } \frac...

Question 10

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

Accepted Answer

A)  $\frac { 11 } { 3 }$ 
B) 8 
C) 4 
D) 16 
E)  $\frac { 11 } { 6 }$
F)2
G)5
H)1 
A)  $\frac { 11 } { 3 }$ 
B) 8 
C) 4 
D) 16 
E)  $\frac { 11 } { 6 }$
F)2
G)5
H)1

Question 11

Evaluate the iterated integral $\int _ { 0 } ^ { x } \int _ { 0 } ^ { \cos y } x \sin y d x d y$ .

Accepted Answer

A)  $\frac { 1 } { 3 }$ 
B)  $\frac { 3 } { 2 }$ 
C)  $\frac { 9 } { 4 }$ 
D) 4 
E)  $\frac { 1 } { 2 }$
F) $\frac { 4 } { 5 }$

G) $\frac { 9 } { 2 }$ 
H)0 
A)  $\frac { 1 } { 3 }$ 
B)  $\frac { 3 } { 2 }$ 
C)  $\frac { 9 } { 4 }$ 
D) 4 
E)  $\frac { 1 } { 2 }$
F) $\frac { 4 } { 5 }$

G) $\frac { 9 } { 2 }$ 
H)0

Question 12

Find the mass of that portion of the solid bounded above by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 3$ which lies in the first octant, if the density varies as the distance from the center of the sphere.

Accepted Answer

The answer of Find the mass of that portion of...

Question 13

Find the moment of inertia $I _ { x }$ of the lamina that occupies the region $D = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ and has density function $p ( x , y ) = x$ .

Accepted Answer

A)  $\frac { 1 } { 24 }$ 
B)  $\frac { 1 } { 18 }$ 
C)  $\frac { 1 } { 12 }$ 
D)  $\frac { 1 } { 9 }$ 
E)  $\frac { 1 } { 6 }$ 
F) $\frac { 1 } { 4 }$

G) $\frac { 1 } { 3 }$

H) $\frac { 1 } { 2 }$ 
A)  $\frac { 1 } { 24 }$ 
B)  $\frac { 1 } { 18 }$ 
C)  $\frac { 1 } { 12 }$ 
D)  $\frac { 1 } { 9 }$ 
E)  $\frac { 1 } { 6 }$ 
F) $\frac { 1 } { 4 }$

G) $\frac { 1 } { 3 }$

H) $\frac { 1 } { 2 }$

Question 14

Find the volume of the region inside the cylinder $x ^ { 2 } + y ^ { 2 } = 7$ which is bounded below by the xy-plane and above by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ .

Accepted Answer

The answer of Find the volume of the region inside...

Question 15

Find the Jacobian of the transformation x = u sin v, y = u cos v when u = 3 and v = 5.

Accepted Answer

A) 3 
B) 5 
C) 7.5 
D) 15 
E) -3
F)-5
G)-7.5
H)-15 
A) 3 
B) 5 
C) 7.5 
D) 15 
E) -3
F)-5
G)-7.5
H)-15

Question 16

Find the Jacobian of the transformation $x = \rho \sin \phi \cos \theta$ , $y = \rho \sin \phi \sin \theta$ , $z = \rho \cos \phi$ .

Accepted Answer

A) $\theta$ 
B)  $\beta ^ { 2 } \sin \phi$ 
C)  $\rho \sin \phi$ 
D)  $\rho ^ { 2 } \cos \phi$ 
E) 2$\theta$
F) $\rho ^ { 2 } \sin \theta$  
G) $\rho \sin \theta$ 
H) $\rho \cos \phi$ 
A) $\theta$ 
B)  $\beta ^ { 2 } \sin \phi$ 
C)  $\rho \sin \phi$ 
D)  $\rho ^ { 2 } \cos \phi$ 
E) 2$\theta$
F) $\rho ^ { 2 } \sin \theta$  
G) $\rho \sin \theta$ 
H) $\rho \cos \phi$

Question 17

Find the area of the surface with vector equation r $( u , v ) = \left\langle u + 2 v , u - 2 v , u ^ { 2 } + 2 v ^ { 2 } \right\rangle$ , $u ^ { 2 } + v ^ { 2 } \leq 4$ .

Accepted Answer

The answer of Find the area of the surface with...

Question 18

Find the polar moment of inertia $I _ { 0 }$ of the lamina that occupies the region $D = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ and has density function $p ( x , y ) = 1$ .

Accepted Answer

A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $\frac { 2 } { 3 }$  
F) $\frac { 3 } { 4 }$

G) $\frac { 5 } { 6 }$

H)1 
A)  $\frac { 1 } { 6 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 3 }$ 
D)  $\frac { 1 } { 2 }$ 
E)  $\frac { 2 } { 3 }$  
F) $\frac { 3 } { 4 }$

G) $\frac { 5 } { 6 }$

H)1

Question 19

Find the center of mass of $D = \left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 4 , x \geq 0 , y \geq 0 \right\}$ if $p ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Accepted Answer

The answer of Find the center of mass of \(D...

Question 20

Find the mass of a solid ball of radius 2 if the density at each point (x, y, z) is $\frac { 3 } { 1 + \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } }$ .

Accepted Answer

The answer of Find the mass of a solid ball...

Evaluate the double integral $\iint _ { R } \frac { y } { ( x + 1 ) ^ { 2 } } d A$ , where $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ .

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

Find the mass of the solid that occupies the region $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ and has density function $\rho ( x , y , z ) = x$ .

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

Find the Jacobian of the transformation $x = r \cos \theta$ , $y = r \sin \theta$ , z = z.

Use a triple integral in spherical coordinates to find the volume of that part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ which lies inside the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Find the area of the portion of the surface $z = x ^ { 2 } + y ^ { 2 }$ inside the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ .

Evaluate $\iint _ { R } \frac { 1 } { x } d A$ , where R is the region bounded by the curves $x ^ { 2 } + y ^ { 2 } = 1$ , $y = x$ , $x = 2$ , and $y = 0$ .

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

Evaluate the iterated integral $\int _ { 0 } ^ { x } \int _ { 0 } ^ { \cos y } x \sin y d x d y$ .

Find the mass of that portion of the solid bounded above by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 3$ which lies in the first octant, if the density varies as the distance from the center of the sphere.

Find the moment of inertia $I _ { x }$ of the lamina that occupies the region $D = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ and has density function $p ( x , y ) = x$ .

Find the volume of the region inside the cylinder $x ^ { 2 } + y ^ { 2 } = 7$ which is bounded below by the xy-plane and above by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ .

Find the Jacobian of the transformation x = u sin v, y = u cos v when u = 3 and v = 5.

Find the Jacobian of the transformation $x = \rho \sin \phi \cos \theta$ , $y = \rho \sin \phi \sin \theta$ , $z = \rho \cos \phi$ .

Find the area of the surface with vector equation r $( u , v ) = \left\langle u + 2 v , u - 2 v , u ^ { 2 } + 2 v ^ { 2 } \right\rangle$ , $u ^ { 2 } + v ^ { 2 } \leq 4$ .

Find the polar moment of inertia $I _ { 0 }$ of the lamina that occupies the region $D = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ and has density function $p ( x , y ) = 1$ .

Find the center of mass of $D = \left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 4 , x \geq 0 , y \geq 0 \right\}$ if $p ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Find the mass of a solid ball of radius 2 if the density at each point (x, y, z) is $\frac { 3 } { 1 + \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } }$ .

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Exam 12: Multiple Integrals

Evaluate the double integral ∬Ry(x+1)2dA\iint _ { R } \frac { y } { ( x + 1 ) ^ { 2 } } d A∬R​(x+1)2y​dA , where R={(x,y)∣0≤x≤1,0≤y≤1}R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}R={(x,y)∣0≤x≤1,0≤y≤1} .

Evaluate the iterated integral ∫01∫01x2ydxdy\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } x ^ { 2 } y d x d y∫01​∫01​x2ydxdy .

Find the mass of the solid that occupies the region 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} and has density function ρ(x,y,z)=x\rho ( x , y , z ) = xρ(x,y,z)=x .

Evaluate the iterated integral ∫0x/2∫yx/2∫0xycos⁡(zy)dzdxdy\int _ { 0 } ^ { x / 2 } \int _ { y } ^ { x / 2 } \int _ { 0 } ^ { x y } \cos \left( \frac { z } { y } \right) d z d x d y∫0x/2​∫yx/2​∫0xy​cos(yz​)dzdxdy .

Find the Jacobian of the transformation x=rcos⁡θx = r \cos \thetax=rcosθ , y=rsin⁡θy = r \sin \thetay=rsinθ , z = z.

Use a triple integral in spherical coordinates to find the volume of that part of the sphere x2+y2+z2=9x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9x2+y2+z2=9 which lies inside the cone z=x2+y2z = \sqrt { x ^ { 2 } + y ^ { 2 } }z=x2+y2​ .

Find the area of the portion of the surface z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 inside the cylinder x2+y2=4x ^ { 2 } + y ^ { 2 } = 4x2+y2=4 .

Evaluate ∬R1xdA\iint _ { R } \frac { 1 } { x } d A∬R​x1​dA , where R is the region bounded by the curves x2+y2=1x ^ { 2 } + y ^ { 2 } = 1x2+y2=1 , y=xy = xy=x , x=2x = 2x=2 , and y=0y = 0y=0 .

Evaluate the iterated integral ∫−11∫02∫01x2+y2+z2dxdydz\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 1 } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } d x d y d z∫−11​∫02​∫01​x2+y2+z2dxdydz .

Evaluate the iterated integral ∫0x∫0cos⁡yxsin⁡ydxdy\int _ { 0 } ^ { x } \int _ { 0 } ^ { \cos y } x \sin y d x d y∫0x​∫0cosy​xsinydxdy .

Find the mass of that portion of the solid bounded above by the sphere x2+y2+z2=3x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 3x2+y2+z2=3 which lies in the first octant, if the density varies as the distance from the center of the sphere.

Find the moment of inertia IxI _ { x }Ix​ of the lamina that occupies the region D={(x,y)∣0≤x≤1,0≤y≤1}D = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}D={(x,y)∣0≤x≤1,0≤y≤1} and has density function p(x,y)=xp ( x , y ) = xp(x,y)=x .

Find the volume of the region inside the cylinder x2+y2=7x ^ { 2 } + y ^ { 2 } = 7x2+y2=7 which is bounded below by the xy-plane and above by the sphere x2+y2+z2=16x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16x2+y2+z2=16 .

Find the Jacobian of the transformation x = u sin v, y = u cos v when u = 3 and v = 5.

Find the Jacobian of the transformation x=ρsin⁡ϕcos⁡θx = \rho \sin \phi \cos \thetax=ρsinϕcosθ , y=ρsin⁡ϕsin⁡θy = \rho \sin \phi \sin \thetay=ρsinϕsinθ , z=ρcos⁡ϕz = \rho \cos \phiz=ρcosϕ .

Find the area of the surface with vector equation r (u,v)=⟨u+2v,u−2v,u2+2v2⟩( u , v ) = \left\langle u + 2 v , u - 2 v , u ^ { 2 } + 2 v ^ { 2 } \right\rangle(u,v)=⟨u+2v,u−2v,u2+2v2⟩ , u2+v2≤4u ^ { 2 } + v ^ { 2 } \leq 4u2+v2≤4 .

Find the polar moment of inertia I0I _ { 0 }I0​ of the lamina that occupies the region D={(x,y)∣0≤x≤1,0≤y≤1}D = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}D={(x,y)∣0≤x≤1,0≤y≤1} and has density function p(x,y)=1p ( x , y ) = 1p(x,y)=1 .

Find the center of mass 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} if p(x,y)=x2+y2p ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }p(x,y)=x2+y2​ .

Find the mass of a solid ball of radius 2 if the density at each point (x, y, z) is 31+x2+y2+z2\frac { 3 } { 1 + \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } }1+x2+y2+z2​3​ .

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Evaluate the double integral $\iint _ { R } \frac { y } { ( x + 1 ) ^ { 2 } } d A$ , where $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ .

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

Find the mass of the solid that occupies the region $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ and has density function $\rho ( x , y , z ) = x$ .

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

Find the Jacobian of the transformation $x = r \cos \theta$ , $y = r \sin \theta$ , z = z.

Use a triple integral in spherical coordinates to find the volume of that part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ which lies inside the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Find the area of the portion of the surface $z = x ^ { 2 } + y ^ { 2 }$ inside the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ .

Evaluate $\iint _ { R } \frac { 1 } { x } d A$ , where R is the region bounded by the curves $x ^ { 2 } + y ^ { 2 } = 1$ , $y = x$ , $x = 2$ , and $y = 0$ .

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

Evaluate the iterated integral $\int _ { 0 } ^ { x } \int _ { 0 } ^ { \cos y } x \sin y d x d y$ .

Find the mass of that portion of the solid bounded above by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 3$ which lies in the first octant, if the density varies as the distance from the center of the sphere.

Find the moment of inertia $I _ { x }$ of the lamina that occupies the region $D = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ and has density function $p ( x , y ) = x$ .

Find the volume of the region inside the cylinder $x ^ { 2 } + y ^ { 2 } = 7$ which is bounded below by the xy-plane and above by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ .

Find the Jacobian of the transformation $x = \rho \sin \phi \cos \theta$ , $y = \rho \sin \phi \sin \theta$ , $z = \rho \cos \phi$ .

Find the area of the surface with vector equation r $( u , v ) = \left\langle u + 2 v , u - 2 v , u ^ { 2 } + 2 v ^ { 2 } \right\rangle$ , $u ^ { 2 } + v ^ { 2 } \leq 4$ .

Find the polar moment of inertia $I _ { 0 }$ of the lamina that occupies the region $D = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ and has density function $p ( x , y ) = 1$ .

Find the center of mass of $D = \left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 4 , x \geq 0 , y \geq 0 \right\}$ if $p ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Find the mass of a solid ball of radius 2 if the density at each point (x, y, z) is $\frac { 3 } { 1 + \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } }$ .