Question 1

The iterated integral $\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { \pi } \int _ { 0 } ^ { 2 } \rho ^ { 2 } \sin ( \varphi ) d \rho d \varphi d \theta$ is

Accepted Answer

A)  $\frac { 64 \pi } { 3 }$ 
B)  $\frac { 32 \pi } { 3 }$ 
C)  $\frac { 16 \pi } { 3 }$ 
D)  $\frac { 8 \pi } { 3 }$ 
E)  $\frac { 4 \pi } { 3 }$ 
A)  $\frac { 64 \pi } { 3 }$ 
B)  $\frac { 32 \pi } { 3 }$ 
C)  $\frac { 16 \pi } { 3 }$ 
D)  $\frac { 8 \pi } { 3 }$ 
E)  $\frac { 4 \pi } { 3 }$

Question 2

The surface area of the surface cut from $x ^ { 2 } + y ^ { 2 } = 9$ by z = x is

Accepted Answer

A)  $6 \sqrt { 8 }$ 
B)  $2 \sqrt { 6 }$ 
C)  $\sqrt { 6 }$ 
D) 18 
E) 9 
A)  $6 \sqrt { 8 }$ 
B)  $2 \sqrt { 6 }$ 
C)  $\sqrt { 6 }$ 
D) 18 
E) 9

Question 3

Let $x = u + 3$ and $y = v - 4 .$ Then $\frac { \partial ( x , y ) } { \partial ( u , v ) }$ is

Accepted Answer

A)  $w$ 
B) 1 
C)  $u$ 
D)  $v$ 
E)  $- w$ 
A)  $w$ 
B) 1 
C)  $u$ 
D)  $v$ 
E)  $- w$

Question 4

The surface area of the surface cut from $x ^ { 2 } + y ^ { 2 } = 25$ by x = 0, x = 1, z = 1, and z = 3 is

Accepted Answer

A)  $10 \sin ^ { - 1 } \left( \frac { 1 } { 5 } \right)$ 
B)  $10 \sin ^ { - 1 } \left( \frac { 2 } { 5 } \right)$ 
C)  $10 \sin ^ { - 1 } \left( \frac { 3 } { 5 } \right)$ 
D)  $10 \sin ^ { - 1 } \left( \frac { 4 } { 5 } \right)$ 
E)  $10 \sin ^ { - 1 } \left( \frac { 6 } { 5 } \right)$ 
A)  $10 \sin ^ { - 1 } \left( \frac { 1 } { 5 } \right)$ 
B)  $10 \sin ^ { - 1 } \left( \frac { 2 } { 5 } \right)$ 
C)  $10 \sin ^ { - 1 } \left( \frac { 3 } { 5 } \right)$ 
D)  $10 \sin ^ { - 1 } \left( \frac { 4 } { 5 } \right)$ 
E)  $10 \sin ^ { - 1 } \left( \frac { 6 } { 5 } \right)$

Question 5

The iterated integral $\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 4 } } \int _ {\frac { \pi } { 4 }  } ^ { \varphi} \int _ { 0 } ^ { \csc \theta } \rho ^ { 3 } \sin ( \varphi ) \sin ^ { 2 } \theta d \rho d \theta d \varphi$  is

Accepted Answer

A)  $\frac { \sqrt { 2 } - 1 } { 2 }$ 
B)  $\frac { \sqrt { 2 } - 1 } { 3 }$ 
C)  $\frac { \sqrt { 2 } - 1 } { 4 }$ 
D)  $\frac { \sqrt { 2 } - 1 } { 5 }$ 
E)  $\frac { \sqrt { 2 } - 1 } { 6 }$ 
A)  $\frac { \sqrt { 2 } - 1 } { 2 }$ 
B)  $\frac { \sqrt { 2 } - 1 } { 3 }$ 
C)  $\frac { \sqrt { 2 } - 1 } { 4 }$ 
D)  $\frac { \sqrt { 2 } - 1 } { 5 }$ 
E)  $\frac { \sqrt { 2 } - 1 } { 6 }$

Question 6

The iterated integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - y ^ { 2 } } } \int _ { 0 } ^ { 2 - y } z d x d z d y$ is

Accepted Answer

A)  $\frac { 10 } { 3 }$ 
B)  $\frac { 3 } { 10 }$ 
C)  $\frac { 10 } { 7 }$ 
D)  $\frac { 10 } { 5 }$ 
E)  $\frac { 7 } { 10 }$ 
A)  $\frac { 10 } { 3 }$ 
B)  $\frac { 3 } { 10 }$ 
C)  $\frac { 10 } { 7 }$ 
D)  $\frac { 10 } { 5 }$ 
E)  $\frac { 7 } { 10 }$

Question 7

The volume of the surface bounded by the graph of $z = 4 x + 2 y$ with $0 \leq x \leq 1,0 \leq y \leq 1$ is

Accepted Answer

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

Question 8

The moment of inertia of a lamina in the shape of a region in the xy-plane bounded by $x ^ { 2 } + y ^ { 2 } = 1 , y \geq 0$ and the x-axis with area density $\rho ( x , y ) = y$ about the x-axis is

Accepted Answer

A) 13 
B) 9 
C)  $\frac { 65 } { 8 }$ 
D)  $\frac { 4 } { 15 }$ 
E)  $\frac { 1 } { 15 }$ 
A) 13 
B) 9 
C)  $\frac { 65 } { 8 }$ 
D)  $\frac { 4 } { 15 }$ 
E)  $\frac { 1 } { 15 }$

Question 9

By Fubini's Theorem, the double integral $\iint$R $\frac { y } { x } d A$ with $R = \{ ( x , y ) : 1 \leq x \leq 2,0 \leq y \leq 2 \}$ is

Accepted Answer

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

Question 10

The cylindrical coordinates of the given rectangular coordinates $\left( - 1,0 , \frac { 1 } { 2 } \right)$ are

Accepted Answer

A)  $\left( 1 , \frac { \pi } { 6 } , \frac { 1 } { 2 } \right)$ 
B)  $\left( 1 , \frac { \pi } { 4 } , \frac { 1 } { 2 } \right)$ 
C)  $\left( 1 , \frac { \pi } { 3 } , \frac { 1 } { 2 } \right)$ 
D)  $\left( 1 , \frac { \pi } { 2 } , \frac { 1 } { 2 } \right)$ 
E)  $\left( 1 , \pi , \frac { 1 } { 2 } \right)$ 
A)  $\left( 1 , \frac { \pi } { 6 } , \frac { 1 } { 2 } \right)$ 
B)  $\left( 1 , \frac { \pi } { 4 } , \frac { 1 } { 2 } \right)$ 
C)  $\left( 1 , \frac { \pi } { 3 } , \frac { 1 } { 2 } \right)$ 
D)  $\left( 1 , \frac { \pi } { 2 } , \frac { 1 } { 2 } \right)$ 
E)  $\left( 1 , \pi , \frac { 1 } { 2 } \right)$

Question 11

By Fubini's Theorem, the double integral $\iint$R $4 x e ^ { y } d A$ with $R = \{ ( x , y ) : 0 \leq x \leq 1,0 \leq y \leq 1 \}$ is

Accepted Answer

A)  $e - 1$ 
B)  $e + 1$ 
C)  $2 ( e - 1 )$ 
D)  $2 ( e + 1 )$ 
E)  $e + 2$ 
A)  $e - 1$ 
B)  $e + 1$ 
C)  $2 ( e - 1 )$ 
D)  $2 ( e + 1 )$ 
E)  $e + 2$

Question 12

The surface area of the surface cut from $x ^ { 2 } + z ^ { 2 } = 16$ by x = 0, x = 2, y = 0, and y = 3 is

Accepted Answer

A)  $10 \sin ^ { - 1 } \left( \frac { 1 } { 5 } \right)$ 
B)  $\frac { \pi } { 2 }$ 
C)  $\pi$ 
D)  $\frac { 3 \pi } { 2 }$ 
E)  $2 \pi$ 
A)  $10 \sin ^ { - 1 } \left( \frac { 1 } { 5 } \right)$ 
B)  $\frac { \pi } { 2 }$ 
C)  $\pi$ 
D)  $\frac { 3 \pi } { 2 }$ 
E)  $2 \pi$

Question 13

The spherical coordinates of the given rectangular coordinates (1, -1, $\sqrt { 2 }$ ) are

Accepted Answer

A)  $\left( 2 , - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$ 
B)  $\left( 2 , \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$ 
C)  $\left( 2 , - \frac { \pi } { 4 } , - \frac { \pi } { 4 } \right)$ 
D)  $\left( 2 , \frac { \pi } { 3 } , - \frac { \pi } { 4 } \right)$ 
E)  $\left( 2 , \frac { \pi } { 6 } , - \frac { \pi } { 4 } \right)$ 
A)  $\left( 2 , - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$ 
B)  $\left( 2 , \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$ 
C)  $\left( 2 , - \frac { \pi } { 4 } , - \frac { \pi } { 4 } \right)$ 
D)  $\left( 2 , \frac { \pi } { 3 } , - \frac { \pi } { 4 } \right)$ 
E)  $\left( 2 , \frac { \pi } { 6 } , - \frac { \pi } { 4 } \right)$

Question 14

The surface area of the surface z = 2xy in the first octant lying inside $4 x ^ { 2 } + 4 y ^ { 2 } = 1$ is

Accepted Answer

A)  $\frac { \pi ( 2 \sqrt { 2 } - 1 ) } { 4 }$ 
B)  $\frac { \pi ( 2 \sqrt { 2 } - 1 ) } { 6 }$ 
C)  $\frac { \pi ( 2 \sqrt { 2 } - 1 ) } { 24 }$ 
D)  $\frac { \pi ( 2 \sqrt { 2 } - 1 ) } { 8 }$ 
E)  $\frac { \pi ( 2 \sqrt { 2 } - 1 ) } { 3 }$ 
A)  $\frac { \pi ( 2 \sqrt { 2 } - 1 ) } { 4 }$ 
B)  $\frac { \pi ( 2 \sqrt { 2 } - 1 ) } { 6 }$ 
C)  $\frac { \pi ( 2 \sqrt { 2 } - 1 ) } { 24 }$ 
D)  $\frac { \pi ( 2 \sqrt { 2 } - 1 ) } { 8 }$ 
E)  $\frac { \pi ( 2 \sqrt { 2 } - 1 ) } { 3 }$

Question 15

The volume of the solid in the first octant bounded by $x + y + 2 z = 2$ and $2 x + 2 y + z = 4$ is

Accepted Answer

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

Question 16

If R is the region bounded by $x = \sqrt { 1 - y ^ { 2 } } , y = x$ and the positive x-axis, then $\iint$R $x d A$  in polar form is

Accepted Answer

A)  $\int _ { 0 } ^ { \frac { \pi } { 4 } } \left[ \int _ { 0 } ^ { 1 } r \cos \theta d r \right] d \theta$ 
B)  $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 1 } r \cos \theta d r \right] d \theta$ 
C)  $\int _ { 0 } ^ { \frac { \pi } { 4 } } \left[ \int _ { 0 } ^ { 1 } r ^ { 2 } \cos \theta d r \right] d \theta$ 
D)  $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 1 } r ^ { 2 } \cos \theta d r \right] d \theta$ 
E)  $\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 1 } r ^ { 2 } \cos \theta d r \right] d \theta$ 
A)  $\int _ { 0 } ^ { \frac { \pi } { 4 } } \left[ \int _ { 0 } ^ { 1 } r \cos \theta d r \right] d \theta$ 
B)  $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 1 } r \cos \theta d r \right] d \theta$ 
C)  $\int _ { 0 } ^ { \frac { \pi } { 4 } } \left[ \int _ { 0 } ^ { 1 } r ^ { 2 } \cos \theta d r \right] d \theta$ 
D)  $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 1 } r ^ { 2 } \cos \theta d r \right] d \theta$ 
E)  $\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 1 } r ^ { 2 } \cos \theta d r \right] d \theta$

Question 17

Using a change of variables, the double integral $\iint$R $\sqrt { ( x - y ) ( x + 4 y ) } d A$ where R is the region bounded by the quadrilateral with vertices (0, 0), (1, 1), (5, 0), and (4, -1) is

Accepted Answer

A)  $\frac { 100 } { 3 }$ 
B)  $- \frac { 100 } { 7 }$ 
C)  $\frac { 100 } { 7 }$ 
D)  $\frac { 100 } { 9 }$ 
E)  $- \frac { 100 } { 9 }$ 
A)  $\frac { 100 } { 3 }$ 
B)  $- \frac { 100 } { 7 }$ 
C)  $\frac { 100 } { 7 }$ 
D)  $\frac { 100 } { 9 }$ 
E)  $- \frac { 100 } { 9 }$

Question 18

The center of mass of a lamina in the shape of a region in the xy-plane bounded by the y-axis, y = x and y = 2 - x with area density $\rho ( x , y ) = 6 x + 3 y + 3$ is

Accepted Answer

A)  $\left( \frac { 7 } { 8 } , \frac { 17 } { 16 } \right)$ 
B)  $\left( \frac { 3 } { 8 } , \frac { 19 } { 16 } \right)$ 
C)  $\left( \frac { 5 } { 8 } , \frac { 17 } { 16 } \right)$ 
D)  $\left( \frac { 3 } { 8 } , \frac { 17 } { 16 } \right)$ 
E)  $\left( \frac { 1 } { 8 } , \frac { 17 } { 16 } \right)$ 
A)  $\left( \frac { 7 } { 8 } , \frac { 17 } { 16 } \right)$ 
B)  $\left( \frac { 3 } { 8 } , \frac { 19 } { 16 } \right)$ 
C)  $\left( \frac { 5 } { 8 } , \frac { 17 } { 16 } \right)$ 
D)  $\left( \frac { 3 } { 8 } , \frac { 17 } { 16 } \right)$ 
E)  $\left( \frac { 1 } { 8 } , \frac { 17 } { 16 } \right)$

Question 19

If R is the region bounded by $y = \sqrt { 9 - x ^ { 2 } }$ in the first quadrant, then $\iint$R $y d A$  in polar form is

Accepted Answer

A)  $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 3 } r \sin \theta d r \right] d \theta$ 
B)  $\int _ { 0 } ^ { \frac { \pi } { 4 } } \left[ \int _ { 0 } ^ { 3 } r \sin \theta d r \right] d \theta$ 
C)  $\int _ { - \frac { \pi } { 4 } } ^ { \frac { \pi } { 4 } } \left[ \int _ { 0 } ^ { 3 } r ^ { 2 } \sin \theta d r \right] d \theta$ 
D)  $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 3 } r ^ { 2 } \sin \theta d r \right] d \theta$ 
E)  $\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 3 } r ^ { 2 } \sin \theta d r \right] d \theta$ 
A)  $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 3 } r \sin \theta d r \right] d \theta$ 
B)  $\int _ { 0 } ^ { \frac { \pi } { 4 } } \left[ \int _ { 0 } ^ { 3 } r \sin \theta d r \right] d \theta$ 
C)  $\int _ { - \frac { \pi } { 4 } } ^ { \frac { \pi } { 4 } } \left[ \int _ { 0 } ^ { 3 } r ^ { 2 } \sin \theta d r \right] d \theta$ 
D)  $\int _ { 0 } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 3 } r ^ { 2 } \sin \theta d r \right] d \theta$ 
E)  $\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 2 } } \left[ \int _ { 0 } ^ { 3 } r ^ { 2 } \sin \theta d r \right] d \theta$

Question 20

The iterated integral $\int _ { 1 } ^ { 2 } \int _ { 0 } ^ { x } \int _ { 1 } ^ {x+ x y } x y d z d y d x$ is

Accepted Answer

A)  $\frac { 9 } { 10 }$ 
B)  $\frac { 7 } { 10 }$ 
C)  $\frac { 7 } { 2 }$ 
D)  $\frac { 189 } { 40 }$ 
E)  $\frac { 3 } { 10 }$ 
A)  $\frac { 9 } { 10 }$ 
B)  $\frac { 7 } { 10 }$ 
C)  $\frac { 7 } { 2 }$ 
D)  $\frac { 189 } { 40 }$ 
E)  $\frac { 3 } { 10 }$

The iterated integral $\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { \pi } \int _ { 0 } ^ { 2 } \rho ^ { 2 } \sin ( \varphi ) d \rho d \varphi d \theta$ is

The surface area of the surface cut from $x ^ { 2 } + y ^ { 2 } = 9$ by z = x is

Let $x = u + 3$ and $y = v - 4 .$ Then $\frac { \partial ( x , y ) } { \partial ( u , v ) }$ is

The surface area of the surface cut from $x ^ { 2 } + y ^ { 2 } = 25$ by x = 0, x = 1, z = 1, and z = 3 is

The iterated integral $\int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 4 } } \int _ {\frac { \pi } { 4 } } ^ { \varphi} \int _ { 0 } ^ { \csc \theta } \rho ^ { 3 } \sin ( \varphi ) \sin ^ { 2 } \theta d \rho d \theta d \varphi$ is

The iterated integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - y ^ { 2 } } } \int _ { 0 } ^ { 2 - y } z d x d z d y$ is

The volume of the surface bounded by the graph of $z = 4 x + 2 y$ with $0 \leq x \leq 1,0 \leq y \leq 1$ is

The moment of inertia of a lamina in the shape of a region in the xy-plane bounded by $x ^ { 2 } + y ^ { 2 } = 1 , y \geq 0$ and the x-axis with area density $\rho ( x , y ) = y$ about the x-axis is

By Fubini's Theorem, the double integral $\iint$ R $\frac { y } { x } d A$ with $R = \{ ( x , y ) : 1 \leq x \leq 2,0 \leq y \leq 2 \}$ is

The cylindrical coordinates of the given rectangular coordinates $\left( - 1,0 , \frac { 1 } { 2 } \right)$ are

By Fubini's Theorem, the double integral $\iint$ R $4 x e ^ { y } d A$ with $R = \{ ( x , y ) : 0 \leq x \leq 1,0 \leq y \leq 1 \}$ is

The surface area of the surface cut from $x ^ { 2 } + z ^ { 2 } = 16$ by x = 0, x = 2, y = 0, and y = 3 is

The spherical coordinates of the given rectangular coordinates (1, -1, $\sqrt { 2 }$ ) are

The surface area of the surface z = 2xy in the first octant lying inside $4 x ^ { 2 } + 4 y ^ { 2 } = 1$ is

The volume of the solid in the first octant bounded by $x + y + 2 z = 2$ and $2 x + 2 y + z = 4$ is

If R is the region bounded by $x = \sqrt { 1 - y ^ { 2 } } , y = x$ and the positive x-axis, then $\iint$ _{R $x d A$} in polar form is

Using a change of variables, the double integral $\iint$ _{R $\sqrt { ( x - y ) ( x + 4 y ) } d A$}where R is the region bounded by the quadrilateral with vertices (0, 0), (1, 1), (5, 0), and (4, -1) is

The center of mass of a lamina in the shape of a region in the xy-plane bounded by the y-axis, y = x and y = 2 - x with area density $\rho ( x , y ) = 6 x + 3 y + 3$ is

If R is the region bounded by $y = \sqrt { 9 - x ^ { 2 } }$ in the first quadrant, then $\iint$ _{R $y d A$} in polar form is

The iterated integral $\int _ { 1 } ^ { 2 } \int _ { 0 } ^ { x } \int _ { 1 } ^ {x+ x y } x y d z d y d x$ is

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Exam 15: Multiple Integrals

The iterated integral ∫02π∫0π∫02ρ2sin⁡(φ)dρdφdθ\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { \pi } \int _ { 0 } ^ { 2 } \rho ^ { 2 } \sin ( \varphi ) d \rho d \varphi d \theta∫02π​∫0π​∫02​ρ2sin(φ)dρdφdθ is

The surface area of the surface cut from x2+y2=9x ^ { 2 } + y ^ { 2 } = 9x2+y2=9 by z = x is

Let x=u+3x = u + 3x=u+3 and y=v−4.y = v - 4 .y=v−4. Then ∂(x,y)∂(u,v)\frac { \partial ( x , y ) } { \partial ( u , v ) }∂(u,v)∂(x,y)​ is

The surface area of the surface cut from x2+y2=25x ^ { 2 } + y ^ { 2 } = 25x2+y2=25 by x = 0, x = 1, z = 1, and z = 3 is

The iterated integral ∫02∫04−y2∫02−yzdxdzdy\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - y ^ { 2 } } } \int _ { 0 } ^ { 2 - y } z d x d z d y∫02​∫04−y2​​∫02−y​zdxdzdy is

The volume of the surface bounded by the graph of z=4x+2yz = 4 x + 2 yz=4x+2y with 0≤x≤1,0≤y≤10 \leq x \leq 1,0 \leq y \leq 10≤x≤1,0≤y≤1 is

The moment of inertia of a lamina in the shape of a region in the xy-plane bounded by x2+y2=1,y≥0x ^ { 2 } + y ^ { 2 } = 1 , y \geq 0x2+y2=1,y≥0 and the x-axis with area density ρ(x,y)=y\rho ( x , y ) = yρ(x,y)=y about the x-axis is

By Fubini's Theorem, the double integral ∬\iint∬ R yxdA\frac { y } { x } d Axy​dA with R={(x,y):1≤x≤2,0≤y≤2}R = \{ ( x , y ) : 1 \leq x \leq 2,0 \leq y \leq 2 \}R={(x,y):1≤x≤2,0≤y≤2} is

The cylindrical coordinates of the given rectangular coordinates (−1,0,12)\left( - 1,0 , \frac { 1 } { 2 } \right)(−1,0,21​) are

By Fubini's Theorem, the double integral ∬\iint∬ R 4xeydA4 x e ^ { y } d A4xeydA with R={(x,y):0≤x≤1,0≤y≤1}R = \{ ( x , y ) : 0 \leq x \leq 1,0 \leq y \leq 1 \}R={(x,y):0≤x≤1,0≤y≤1} is

The surface area of the surface cut from x2+z2=16x ^ { 2 } + z ^ { 2 } = 16x2+z2=16 by x = 0, x = 2, y = 0, and y = 3 is

The spherical coordinates of the given rectangular coordinates (1, -1, 2\sqrt { 2 }2​ ) are

The surface area of the surface z = 2xy in the first octant lying inside 4x2+4y2=14 x ^ { 2 } + 4 y ^ { 2 } = 14x2+4y2=1 is

The volume of the solid in the first octant bounded by x+y+2z=2x + y + 2 z = 2x+y+2z=2 and 2x+2y+z=42 x + 2 y + z = 42x+2y+z=4 is

If R is the region bounded by x=1−y2,y=xx = \sqrt { 1 - y ^ { 2 } } , y = xx=1−y2​,y=x and the positive x-axis, then ∬\iint∬ R xdAx d AxdA in polar form is

Using a change of variables, the double integral ∬\iint∬ R (x−y)(x+4y)dA\sqrt { ( x - y ) ( x + 4 y ) } d A(x−y)(x+4y)​dA where R is the region bounded by the quadrilateral with vertices (0, 0), (1, 1), (5, 0), and (4, -1) is

The center of mass of a lamina in the shape of a region in the xy-plane bounded by the y-axis, y = x and y = 2 - x with area density ρ(x,y)=6x+3y+3\rho ( x , y ) = 6 x + 3 y + 3ρ(x,y)=6x+3y+3 is

If R is the region bounded by y=9−x2y = \sqrt { 9 - x ^ { 2 } }y=9−x2​ in the first quadrant, then ∬\iint∬ R ydAy d AydA in polar form is

The iterated integral ∫12∫0x∫1x+xyxydzdydx\int _ { 1 } ^ { 2 } \int _ { 0 } ^ { x } \int _ { 1 } ^ {x+ x y } x y d z d y d x∫12​∫0x​∫1x+xy​xydzdydx is

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

Applications of the Derivative

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Vector Calculus

Differential Equations

Filters

The iterated integral $\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { \pi } \int _ { 0 } ^ { 2 } \rho ^ { 2 } \sin ( \varphi ) d \rho d \varphi d \theta$ is

The surface area of the surface cut from $x ^ { 2 } + y ^ { 2 } = 9$ by z = x is

Let $x = u + 3$ and $y = v - 4 .$ Then $\frac { \partial ( x , y ) } { \partial ( u , v ) }$ is

The surface area of the surface cut from $x ^ { 2 } + y ^ { 2 } = 25$ by x = 0, x = 1, z = 1, and z = 3 is

The iterated integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - y ^ { 2 } } } \int _ { 0 } ^ { 2 - y } z d x d z d y$ is

The volume of the surface bounded by the graph of $z = 4 x + 2 y$ with $0 \leq x \leq 1,0 \leq y \leq 1$ is

The moment of inertia of a lamina in the shape of a region in the xy-plane bounded by $x ^ { 2 } + y ^ { 2 } = 1 , y \geq 0$ and the x-axis with area density $\rho ( x , y ) = y$ about the x-axis is

By Fubini's Theorem, the double integral $\iint$ R $\frac { y } { x } d A$ with $R = \{ ( x , y ) : 1 \leq x \leq 2,0 \leq y \leq 2 \}$ is

The cylindrical coordinates of the given rectangular coordinates $\left( - 1,0 , \frac { 1 } { 2 } \right)$ are

By Fubini's Theorem, the double integral $\iint$ R $4 x e ^ { y } d A$ with $R = \{ ( x , y ) : 0 \leq x \leq 1,0 \leq y \leq 1 \}$ is

The surface area of the surface cut from $x ^ { 2 } + z ^ { 2 } = 16$ by x = 0, x = 2, y = 0, and y = 3 is

The spherical coordinates of the given rectangular coordinates (1, -1, $\sqrt { 2 }$ ) are

The surface area of the surface z = 2xy in the first octant lying inside $4 x ^ { 2 } + 4 y ^ { 2 } = 1$ is

The volume of the solid in the first octant bounded by $x + y + 2 z = 2$ and $2 x + 2 y + z = 4$ is

If R is the region bounded by $x = \sqrt { 1 - y ^ { 2 } } , y = x$ and the positive x-axis, then $\iint$ _{R $x d A$} in polar form is

Using a change of variables, the double integral $\iint$ _{R $\sqrt { ( x - y ) ( x + 4 y ) } d A$}where R is the region bounded by the quadrilateral with vertices (0, 0), (1, 1), (5, 0), and (4, -1) is

The center of mass of a lamina in the shape of a region in the xy-plane bounded by the y-axis, y = x and y = 2 - x with area density $\rho ( x , y ) = 6 x + 3 y + 3$ is

If R is the region bounded by $y = \sqrt { 9 - x ^ { 2 } }$ in the first quadrant, then $\iint$ _{R $y d A$} in polar form is

The iterated integral $\int _ { 1 } ^ { 2 } \int _ { 0 } ^ { x } \int _ { 1 } ^ {x+ x y } x y d z d y d x$ is