Question 1

Let and $x = u + v$ Then $y = u - v$ is

Accepted Answer

A)  $w$ 
B)  $u$ 
C)  $v$ 
D) 2 
E)  $- 2$ 
A)  $w$ 
B)  $u$ 
C)  $v$ 
D) 2 
E)  $- 2$ D

Question 2

The spherical coordinates of the given rectangular coordinates (1, $\sqrt { 3 }$ , 2) are

Accepted Answer

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

Question 3

The iterated integral $\int _ { \frac { \pi } { 2 } } ^ { \pi } \left[ \int _ { 0 } ^ { x } \sin ( 4 x - y ) d y \right] d x$ is

Accepted Answer

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

Question 4

The volume of the solid bounded by $z ^ { 2 } + r ^ { 2 } = 9$ is

Accepted Answer

A) 9? 
B) 81? 
C) 18? 
D) 24? 
E) 36? 
A) 9? 
B) 81? 
C) 18? 
D) 24? 
E) 36?

Question 5

The iterated integral $\int_{0}^{\frac{\pi}{2}} \int_{0}^{\pi} \int_{0}^{2}e^{\rho^{3}}  \rho^{2} d \rho d \theta d \varphi$ is

Accepted Answer

A)  $\frac { \pi ^ { 2 } \left( e ^ { 8 } - 1 \right) } { 3 }$ 
B)  $\frac { \pi ^ { 2 } \left( 1 - e ^ { - 8 } \right) } { 4 }$ 
C)  $\frac { \pi ^ { 2 } \left( e ^ { 8 } - 1 \right) } { 5 }$ 
D)  $\frac { \pi ^ { 2 } \left( e ^ { 8 } - 1 \right) } { 6 }$ 
E)  $\frac { \pi ^ { 2 } \left( e ^ { 8 } - 1 \right) } { 7 }$ 
A)  $\frac { \pi ^ { 2 } \left( e ^ { 8 } - 1 \right) } { 3 }$ 
B)  $\frac { \pi ^ { 2 } \left( 1 - e ^ { - 8 } \right) } { 4 }$ 
C)  $\frac { \pi ^ { 2 } \left( e ^ { 8 } - 1 \right) } { 5 }$ 
D)  $\frac { \pi ^ { 2 } \left( e ^ { 8 } - 1 \right) } { 6 }$ 
E)  $\frac { \pi ^ { 2 } \left( e ^ { 8 } - 1 \right) } { 7 }$

Question 6

The volume of the surface bounded by the graph of $z = 6 x + 4 y$ with $0 \leq x \leq 1,0 \leq y \leq 1$ is

Accepted Answer

A) 2 
B) 3 
C) 4 
D) 5 
E) 7 
A) 2 
B) 3 
C) 4 
D) 5 
E) 7

Question 7

The surface area of the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ above z = 0 and inside $x ^ { 2 } + y ^ { 2 } = x$ is

Accepted Answer

A)  $\frac { \pi + 2 } { 2 }$ 
B)  $\frac { \pi - 2 } { 2 }$ 
C)  $\pi + 2$ 
D)  $\pi - 2$ 
E)  $2 \pi - 2$ 
A)  $\frac { \pi + 2 } { 2 }$ 
B)  $\frac { \pi - 2 } { 2 }$ 
C)  $\pi + 2$ 
D)  $\pi - 2$ 
E)  $2 \pi - 2$

Question 8

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

Accepted Answer

A)  $\left( - \frac { 6 } { 5 } , \frac { 9 } { 5 } \right)$ 
B)  $\left( \frac { 6 } { 5 } , - \frac { 6 } { 5 } \right)$ 
C)  $\left( \frac { 6 } { 5 } , - \frac { 9 } { 5 } \right)$ 
D)  $\left( \frac { 6 } { 5 } , \frac { 6 } { 5 } \right)$ 
E)  $\left( \frac { 6 } { 5 } , \frac { 9 } { 5 } \right)$ 
A)  $\left( - \frac { 6 } { 5 } , \frac { 9 } { 5 } \right)$ 
B)  $\left( \frac { 6 } { 5 } , - \frac { 6 } { 5 } \right)$ 
C)  $\left( \frac { 6 } { 5 } , - \frac { 9 } { 5 } \right)$ 
D)  $\left( \frac { 6 } { 5 } , \frac { 6 } { 5 } \right)$ 
E)  $\left( \frac { 6 } { 5 } , \frac { 9 } { 5 } \right)$

Question 9

The volume of the solid in the first octant bounded by and all the coordinate planes is

Accepted Answer

A)  $\frac { 8 \sqrt { 2 } } { 3 }$ 
B)  $\frac { 5 \sqrt { 2 } } { 3 }$ 
C)  $\frac { 4 \sqrt { 2 } } { 3 }$ 
D)  $\frac { 2 \sqrt { 2 } } { 3 }$ 
E)  $\frac { \sqrt { 2 } } { 3 }$ 
A)  $\frac { 8 \sqrt { 2 } } { 3 }$ 
B)  $\frac { 5 \sqrt { 2 } } { 3 }$ 
C)  $\frac { 4 \sqrt { 2 } } { 3 }$ 
D)  $\frac { 2 \sqrt { 2 } } { 3 }$ 
E)  $\frac { \sqrt { 2 } } { 3 }$

Question 10

The iterated integral $\int _ { - 1 } ^ { 1 } \left[ \int _ { 1 } ^ { e ^ { z } } \frac { x } { y } d y \right] d x$ is

Accepted Answer

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

Question 11

If $\sigma ( x , y , z ) = k$ is the volume density of the solid inside $x ^ { 2 } + y ^ { 2 } - 2 x = 0$ below $x ^ { 2 } + y ^ { 2 } = z ^ { 2 }$ and above z = 0 then its moment of inertia about the z-axis is

Accepted Answer

A)  $\frac { 512 k } { 75 }$ 
B)  $\frac { 256 k } { 75 }$ 
C)  $\frac { 256 k } { 15 }$ 
D)  $\frac { 512 k } { 15 }$ 
E)  $\frac { 128 k } { 15 }$ 
A)  $\frac { 512 k } { 75 }$ 
B)  $\frac { 256 k } { 75 }$ 
C)  $\frac { 256 k } { 15 }$ 
D)  $\frac { 512 k } { 15 }$ 
E)  $\frac { 128 k } { 15 }$

Question 12

The moment of inertia of a lamina in the shape of a region in the xy-plane bounded by $y = \frac { 3 } { 4 } x , x = 4$ and the x-axis with area density $\rho ( x , y ) = k$ about the x-axis is

Accepted Answer

A)  $13 k$ 
B)  $9 k$ 
C)  $8 k$ 
D)  $11 k$ 
E)  $7 k$ 
A)  $13 k$ 
B)  $9 k$ 
C)  $8 k$ 
D)  $11 k$ 
E)  $7 k$

Question 13

The rectangular coordinates of the given spherical coordinates $\left( 1 , \frac { \pi } { 4 } , \frac { \pi } { 6 } \right)$ are

Accepted Answer

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

Question 14

If R is the region bounded by y = x, x = ? and the x-axis, then $\iint$R $\cos ( x + y ) d A$ is

Accepted Answer

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

Question 15

The volume of the solid bounded $x = 0 , x = 1 - y ^ { 2 } , x ^ { 2 } + y ^ { 2 } = z$ and z = 0 is

Accepted Answer

A)  $\frac { 3 } { 7 }$ 
B)  $\frac { 4 } { 7 }$ 
C)  $\frac { 5 } { 7 }$ 
D)  $\frac { 6 } { 7 }$ 
E)  $\frac { 8 } { 7 }$ 
A)  $\frac { 3 } { 7 }$ 
B)  $\frac { 4 } { 7 }$ 
C)  $\frac { 5 } { 7 }$ 
D)  $\frac { 6 } { 7 }$ 
E)  $\frac { 8 } { 7 }$

Question 16

Let $x = u ^ { 3 } - 3 u v ^ { 2 }$ and $y = 3 u ^ { 2 } v - v ^ { 3 }$ Then $\frac { \partial ( x , y ) } { \partial ( u , v ) }$ is

Accepted Answer

A)  $9 \left( u ^ { 2 } + v ^ { 2 } \right) ^ { 2 }$ 
B)  $9 \left( u ^ { 2 } - v ^ { 2 } \right) ^ { 2 }$ 
C)  $- 9 \left( u ^ { 2 } + v ^ { 2 } \right) ^ { 2 }$ 
D)  $- 9 \left( u ^ { 2 } - v ^ { 2 } \right) ^ { 2 }$ 
E)  $9 \left( u ^ { 2 } + v ^ { 2 } \right)$ 
A)  $9 \left( u ^ { 2 } + v ^ { 2 } \right) ^ { 2 }$ 
B)  $9 \left( u ^ { 2 } - v ^ { 2 } \right) ^ { 2 }$ 
C)  $- 9 \left( u ^ { 2 } + v ^ { 2 } \right) ^ { 2 }$ 
D)  $- 9 \left( u ^ { 2 } - v ^ { 2 } \right) ^ { 2 }$ 
E)  $9 \left( u ^ { 2 } + v ^ { 2 } \right)$

Question 17

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

Accepted Answer

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

Question 18

Let $x = e ^ { u } \cos v$ and $y = e ^ { u } \sin v$ Then $\frac { \partial ( x , y ) } { \partial ( u , v ) }$ is

Accepted Answer

A)  $u e ^ { u }$ 
B)  $\boldsymbol { u }$ 
C)  $u$ 
D)  $e ^ { 2 u }$ 
E)  $- e ^ { 2 u }$ 
A)  $u e ^ { u }$ 
B)  $\boldsymbol { u }$ 
C)  $u$ 
D)  $e ^ { 2 u }$ 
E)  $- e ^ { 2 u }$

Question 19

If the order of integration of $\int _ { 0 } ^ { 2 } \left[ \int _ { y ^ { 2 } } ^ { 2 y } f ( x , y ) d x \right] d y$ is switched, the result is

Accepted Answer

A)  $\int _ { 0 } ^ { 4 } \left[ \int _ { \frac { x } { 2 } } ^ { \sqrt { x } } f ( x , y ) d y \right] d x$ 
B)  $\int _ { 0 } ^ { 2 } \left[ \int _ { \frac { x } { 2 } } ^ { \sqrt { x } } f ( x , y ) d y \right] d x$ 
C)  $\int _ { 0 } ^ { 4 } \left[ \int _ { \sqrt { x } } ^ { \frac { x } { 2 } } f ( x , y ) d y \right] d x$ 
D)  $\int _ { 0 } ^ { 2 } \left[ \int _ { \sqrt { x } } ^ { \frac { x } { 2 } } f ( x , y ) d y \right] d x$ 
E)  $\int _ { 0 } ^ { 1 } \left[ \int _ { \frac { x } { 2 } } ^ { \sqrt { x } } f ( x , y ) d y \right] d x$ 
A)  $\int _ { 0 } ^ { 4 } \left[ \int _ { \frac { x } { 2 } } ^ { \sqrt { x } } f ( x , y ) d y \right] d x$ 
B)  $\int _ { 0 } ^ { 2 } \left[ \int _ { \frac { x } { 2 } } ^ { \sqrt { x } } f ( x , y ) d y \right] d x$ 
C)  $\int _ { 0 } ^ { 4 } \left[ \int _ { \sqrt { x } } ^ { \frac { x } { 2 } } f ( x , y ) d y \right] d x$ 
D)  $\int _ { 0 } ^ { 2 } \left[ \int _ { \sqrt { x } } ^ { \frac { x } { 2 } } f ( x , y ) d y \right] d x$ 
E)  $\int _ { 0 } ^ { 1 } \left[ \int _ { \frac { x } { 2 } } ^ { \sqrt { x } } f ( x , y ) d y \right] d x$

Question 20

Let and $x = u + v + w$ Then $y = u + v - w$ is

Accepted Answer

A)  $- \mathcal { U }$ 
B)  $- v$ 
C)  $- 1$ 
D) 4 
E)  $- 4$ 
A)  $- \mathcal { U }$ 
B)  $- v$ 
C)  $- 1$ 
D) 4 
E)  $- 4$

Let and $x = u + v$ Then $y = u - v$ is

The spherical coordinates of the given rectangular coordinates (1, $\sqrt { 3 }$ , 2) are

The iterated integral $\int _ { \frac { \pi } { 2 } } ^ { \pi } \left[ \int _ { 0 } ^ { x } \sin ( 4 x - y ) d y \right] d x$ is

The volume of the solid bounded by $z ^ { 2 } + r ^ { 2 } = 9$ is

The iterated integral $\int_{0}^{\frac{\pi}{2}} \int_{0}^{\pi} \int_{0}^{2}e^{\rho^{3}} \rho^{2} d \rho d \theta d \varphi$ is

The volume of the surface bounded by the graph of $z = 6 x + 4 y$ with $0 \leq x \leq 1,0 \leq y \leq 1$ is

The surface area of the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ above z = 0 and inside $x ^ { 2 } + y ^ { 2 } = x$ is

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

The volume of the solid in the first octant bounded by and all the coordinate planes is

The iterated integral $\int _ { - 1 } ^ { 1 } \left[ \int _ { 1 } ^ { e ^ { z } } \frac { x } { y } d y \right] d x$ is

If $\sigma ( x , y , z ) = k$ is the volume density of the solid inside $x ^ { 2 } + y ^ { 2 } - 2 x = 0$ below $x ^ { 2 } + y ^ { 2 } = z ^ { 2 }$ and above z = 0 then its moment of inertia about the z-axis is

The moment of inertia of a lamina in the shape of a region in the xy-plane bounded by $y = \frac { 3 } { 4 } x , x = 4$ and the x-axis with area density $\rho ( x , y ) = k$ about the x-axis is

The rectangular coordinates of the given spherical coordinates $\left( 1 , \frac { \pi } { 4 } , \frac { \pi } { 6 } \right)$ are

If R is the region bounded by y = x, x = ? and the x-axis, then $\iint$ R $\cos ( x + y ) d A$ is

The volume of the solid bounded $x = 0 , x = 1 - y ^ { 2 } , x ^ { 2 } + y ^ { 2 } = z$ and z = 0 is

Let $x = u ^ { 3 } - 3 u v ^ { 2 }$ and $y = 3 u ^ { 2 } v - v ^ { 3 }$ Then $\frac { \partial ( x , y ) } { \partial ( u , v ) }$ is

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

Let $x = e ^ { u } \cos v$ and $y = e ^ { u } \sin v$ Then $\frac { \partial ( x , y ) } { \partial ( u , v ) }$ is

If the order of integration of $\int _ { 0 } ^ { 2 } \left[ \int _ { y ^ { 2 } } ^ { 2 y } f ( x , y ) d x \right] d y$ is switched, the result is

Let and $x = u + v + w$ Then $y = u + v - w$ is

Preparing for Calculus

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Exam 15: Multiple Integrals

Let and x=u+vx = u + vx=u+v Then y=u−vy = u - vy=u−v is

The spherical coordinates of the given rectangular coordinates (1, 3\sqrt { 3 }3​ , 2) are

The iterated integral ∫π2π[∫0xsin⁡(4x−y)dy]dx\int _ { \frac { \pi } { 2 } } ^ { \pi } \left[ \int _ { 0 } ^ { x } \sin ( 4 x - y ) d y \right] d x∫2π​π​[∫0x​sin(4x−y)dy]dx is

The volume of the solid bounded by z2+r2=9z ^ { 2 } + r ^ { 2 } = 9z2+r2=9 is

The iterated integral ∫0π2∫0π∫02eρ3ρ2dρdθdφ\int_{0}^{\frac{\pi}{2}} \int_{0}^{\pi} \int_{0}^{2}e^{\rho^{3}} \rho^{2} d \rho d \theta d \varphi∫02π​​∫0π​∫02​eρ3ρ2dρdθdφ is

The volume of the surface bounded by the graph of z=6x+4yz = 6 x + 4 yz=6x+4y with 0≤x≤1,0≤y≤10 \leq x \leq 1,0 \leq y \leq 10≤x≤1,0≤y≤1 is

The surface area of the surface x2+y2+z2=1x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1x2+y2+z2=1 above z = 0 and inside x2+y2=xx ^ { 2 } + y ^ { 2 } = xx2+y2=x is

The center of mass of a lamina in the shape of a region in the xy-plane bounded by the x-axis, ρ(x,y)=y2\rho ( x , y ) = y ^ { 2 }ρ(x,y)=y2 and the y-axis with area density ρ(x,y)=y2\rho ( x , y ) = y ^ { 2 }ρ(x,y)=y2 is

The volume of the solid in the first octant bounded by and all the coordinate planes is

The iterated integral ∫−11[∫1ezxydy]dx\int _ { - 1 } ^ { 1 } \left[ \int _ { 1 } ^ { e ^ { z } } \frac { x } { y } d y \right] d x∫−11​[∫1ez​yx​dy]dx is

If σ(x,y,z)=k\sigma ( x , y , z ) = kσ(x,y,z)=k is the volume density of the solid inside x2+y2−2x=0x ^ { 2 } + y ^ { 2 } - 2 x = 0x2+y2−2x=0 below x2+y2=z2x ^ { 2 } + y ^ { 2 } = z ^ { 2 }x2+y2=z2 and above z = 0 then its moment of inertia about the z-axis is

The moment of inertia of a lamina in the shape of a region in the xy-plane bounded by y=34x,x=4y = \frac { 3 } { 4 } x , x = 4y=43​x,x=4 and the x-axis with area density ρ(x,y)=k\rho ( x , y ) = kρ(x,y)=k about the x-axis is

The rectangular coordinates of the given spherical coordinates (1,π4,π6)\left( 1 , \frac { \pi } { 4 } , \frac { \pi } { 6 } \right)(1,4π​,6π​) are

If R is the region bounded by y = x, x = ? and the x-axis, then ∬\iint∬ R cos⁡(x+y)dA\cos ( x + y ) d Acos(x+y)dA is

The volume of the solid bounded x=0,x=1−y2,x2+y2=zx = 0 , x = 1 - y ^ { 2 } , x ^ { 2 } + y ^ { 2 } = zx=0,x=1−y2,x2+y2=z and z = 0 is

Let x=u3−3uv2x = u ^ { 3 } - 3 u v ^ { 2 }x=u3−3uv2 and y=3u2v−v3y = 3 u ^ { 2 } v - v ^ { 3 }y=3u2v−v3 Then ∂(x,y)∂(u,v)\frac { \partial ( x , y ) } { \partial ( u , v ) }∂(u,v)∂(x,y)​ is

By Fubini's Theorem, the double integral ∬\iint∬ R yx2dA\frac { y } { x ^ { 2 } } d Ax2y​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

Let x=eucos⁡vx = e ^ { u } \cos vx=eucosv and y=eusin⁡vy = e ^ { u } \sin vy=eusinv Then ∂(x,y)∂(u,v)\frac { \partial ( x , y ) } { \partial ( u , v ) }∂(u,v)∂(x,y)​ is

If the order of integration of ∫02[∫y22yf(x,y)dx]dy\int _ { 0 } ^ { 2 } \left[ \int _ { y ^ { 2 } } ^ { 2 y } f ( x , y ) d x \right] d y∫02​[∫y22y​f(x,y)dx]dy is switched, the result is

Let and x=u+v+wx = u + v + wx=u+v+w Then y=u+v−wy = u + v - wy=u+v−w 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

Let and $x = u + v$ Then $y = u - v$ is

The spherical coordinates of the given rectangular coordinates (1, $\sqrt { 3 }$ , 2) are

The iterated integral $\int _ { \frac { \pi } { 2 } } ^ { \pi } \left[ \int _ { 0 } ^ { x } \sin ( 4 x - y ) d y \right] d x$ is

The volume of the solid bounded by $z ^ { 2 } + r ^ { 2 } = 9$ is

The iterated integral $\int_{0}^{\frac{\pi}{2}} \int_{0}^{\pi} \int_{0}^{2}e^{\rho^{3}} \rho^{2} d \rho d \theta d \varphi$ is

The volume of the surface bounded by the graph of $z = 6 x + 4 y$ with $0 \leq x \leq 1,0 \leq y \leq 1$ is

The surface area of the surface $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ above z = 0 and inside $x ^ { 2 } + y ^ { 2 } = x$ is

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

The iterated integral $\int _ { - 1 } ^ { 1 } \left[ \int _ { 1 } ^ { e ^ { z } } \frac { x } { y } d y \right] d x$ is

If $\sigma ( x , y , z ) = k$ is the volume density of the solid inside $x ^ { 2 } + y ^ { 2 } - 2 x = 0$ below $x ^ { 2 } + y ^ { 2 } = z ^ { 2 }$ and above z = 0 then its moment of inertia about the z-axis is

The moment of inertia of a lamina in the shape of a region in the xy-plane bounded by $y = \frac { 3 } { 4 } x , x = 4$ and the x-axis with area density $\rho ( x , y ) = k$ about the x-axis is

The rectangular coordinates of the given spherical coordinates $\left( 1 , \frac { \pi } { 4 } , \frac { \pi } { 6 } \right)$ are

If R is the region bounded by y = x, x = ? and the x-axis, then $\iint$ R $\cos ( x + y ) d A$ is

The volume of the solid bounded $x = 0 , x = 1 - y ^ { 2 } , x ^ { 2 } + y ^ { 2 } = z$ and z = 0 is

Let $x = u ^ { 3 } - 3 u v ^ { 2 }$ and $y = 3 u ^ { 2 } v - v ^ { 3 }$ Then $\frac { \partial ( x , y ) } { \partial ( u , v ) }$ is

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

Let $x = e ^ { u } \cos v$ and $y = e ^ { u } \sin v$ Then $\frac { \partial ( x , y ) } { \partial ( u , v ) }$ is

If the order of integration of $\int _ { 0 } ^ { 2 } \left[ \int _ { y ^ { 2 } } ^ { 2 y } f ( x , y ) d x \right] d y$ is switched, the result is

Let and $x = u + v + w$ Then $y = u + v - w$ is