Question 1

Evaluate the iterated integral.
$$\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 10 x y d x d y$$

Accepted Answer

The answer of Evaluate the iterated integral.
\[\int _ { 1...

Question 2

$$\text { Identify the surface with equation } \phi = \frac { \pi } { 6 }$$

Accepted Answer

Upper half of a righ

Question 3

Use spherical coordinates to evaluate $\iiint _ { B } \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d V$, where $B$ is the ball $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 7$.
Select the correct answer.

Accepted Answer

A)  $2401 \pi$ 
B)  $343 \pi$ 
C)  $686 \pi$ 
D)  $7 \pi$ 
A)  $2401 \pi$ 
B)  $343 \pi$ 
C)  $686 \pi$ 
D)  $7 \pi$

Question 4

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 . Select the correct answer.

Accepted Answer

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

Question 5

Find the area of the surface $S$ where $S$ is the part of the plane $z = 3 x ^ { 2 } + y$ that lies above the triangular region with vertices $( 0,0 ) , ( 2,0 )$, and $( 2,2 )$.

Accepted Answer

The answer of Find the area of the surface $S$...

Question 6

Find the mass and the center of mass of the lamina occupying the region $R$, where $R$ is the triangular region with vertices $( 0,0 ) , ( 3,8 )$, and $( 6,0 )$, and having the mass density $\rho ( x , y ) = x$.

Accepted Answer

The answer of Find the mass and the center of...

Question 7

Use the Midpoint Rule with four squares of equal size to estimate the double integral.
$$\iint _ { R } \cos \left( x ^ { 4 } + y ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 0.4,0 \leq y \leq 0.4 \}$$

Accepted Answer

The answer of Use the Midpoint Rule with four squares...

Question 8

Evaluate the double integral by first identifying it as the volume of a solid.
$$\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 4 \leq x \leq 7,2 \leq y \leq 6 \}$$
Select the correct answer.

Accepted Answer

A)  $- 120$ 
B)  280 
C)  180 
D)  $- 20$ 
E)  80 
A)  $- 120$ 
B)  280 
C)  180 
D)  $- 20$ 
E)  80

Question 9

An agricultural sprinkler distributes water in a circular pattern of radius $100 \mathrm { ft }$. It supplies water to a depth of $e ^ { - r }$ feet per hour at a distance of $r$ feet from the sprinkler. What is the total amount of water supplied per hour to the region inside the circle of radius 15 feet centered at the sprinkler?

Accepted Answer

The answer of An agricultural sprinkler distributes water in a...

Question 10

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

Accepted Answer

The answer of Use polar coordinates to find the volume...

Question 11

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

Accepted Answer

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) 
$m = 20 , ( \bar { x } , \bar { y } ) = \left( \frac { 5 } { 3 } , \frac { 7 } { 3 } \right)$ 
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 12

Calculate the double integral.
Select the correct answer.
$$\iint _ { R } \left( 15 x ^ { 2 } y ^ { 3 } - 25 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}$$

Accepted Answer

A)  350 
B)  300 
C)  310 
D)  370 
E)  330 
A)  350 
B)  300 
C)  310 
D)  370 
E)  330

Question 13

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 $x y = 2 , x y = 4 ; x = \frac { u } { v } , y = v$.
Select the correct answer.

Accepted Answer

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

Question 14

Find the Jacobian of the transformation.
$$x = 5 \alpha \sin \beta , y = 4 \alpha \cos \beta$$

Accepted Answer

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

Question 15

Evaluate the integral $\iiint _ { B } f ( x , y , z ) d V$ where $f ( x , y , z ) = x y ^ { 2 } + y z ^ { 2 }$ and $B = \{ ( x , y , z ) \mid 0 \leq x \leq 2 , - 1 \leq y \leq 1,0 \leq z \leq 3 \}$ with respect to $x , y$, and $z$, in that order.

Accepted Answer

A)  28 
B)  36 
C)  4 
D)  24 
A)  28 
B)  36 
C)  4 
D)  24

Question 16

Use the transformation $x = \sqrt { 2 } u - \sqrt { \frac { 2 } { 3 } } v , y = \sqrt { 2 } u + \sqrt { \frac { 2 } { 3 } } v$ 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 $x ^ { 2 } - x y + y ^ { 2 } = 2$.

Accepted Answer

A)  $\frac { 100 \pi } { 3 ^ { 3 / 2 } }$ 
B)  $\frac { 25 } { \sqrt { 3 } }$ 
C)  $\frac { 5 \pi } { \sqrt { 3 } }$ 
D)  $\frac { 5 \pi ^ { 2 } } { \sqrt { 3 } }$ 
E)  $\frac { 4 \pi } { \sqrt { 3 } }$ 
A)  $\frac { 100 \pi } { 3 ^ { 3 / 2 } }$ 
B)  $\frac { 25 } { \sqrt { 3 } }$ 
C)  $\frac { 5 \pi } { \sqrt { 3 } }$ 
D)  $\frac { 5 \pi ^ { 2 } } { \sqrt { 3 } }$ 
E)  $\frac { 4 \pi } { \sqrt { 3 } }$

Question 17

Calculate the iterated integral. Round your answer to two decimal places.
$$\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { 3 } \sqrt { 25 x + 25 y } d x d y$$

Accepted Answer

The answer of Calculate the iterated integral. Round your answer...

Question 18

Calculate the iterated integral.
$$\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } 5 \cos \left( y ^ { 2 } \right) d x d y$$

Accepted Answer

The answer of Calculate the iterated integral.
\[\int _ { 0...

Question 19

Evaluate the integral by reversing the order of integration.
$$\int _ { 0 } ^ { 1 } \int _ { 2 n } ^ { 2 } e ^ { x ^ { 2 } } d x d y$$

Accepted Answer

The answer of Evaluate the integral by reversing the order...

Question 20

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 $x ^ { 2 } + y ^ { 2 } = 36$.

Accepted Answer

A)  $( 1297 \sqrt { 1297 } + 5 \sqrt { 5 } ) \pi$ 
B)  $( 1297 \sqrt { 1297 } - 6 \sqrt { 5 } ) \pi$ 
C)  $\frac { 1 } { 9 } ( 1297 \sqrt { 1297 } - 6 \sqrt { 5 } ) \pi$ 
D)  $\frac { 1 } { 9 } ( 1297 \sqrt { 1297 } - 5 \sqrt { 5 } )$ 
E)  $\frac { 1 } { 9 } ( 1297 \sqrt { 1297 } - 5 \sqrt { 5 } ) \pi$ 
A)  $( 1297 \sqrt { 1297 } + 5 \sqrt { 5 } ) \pi$ 
B)  $( 1297 \sqrt { 1297 } - 6 \sqrt { 5 } ) \pi$ 
C)  $\frac { 1 } { 9 } ( 1297 \sqrt { 1297 } - 6 \sqrt { 5 } ) \pi$ 
D)  $\frac { 1 } { 9 } ( 1297 \sqrt { 1297 } - 5 \sqrt { 5 } )$ 
E)  $\frac { 1 } { 9 } ( 1297 \sqrt { 1297 } - 5 \sqrt { 5 } ) \pi$

Evaluate the iterated integral. $\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 10 x y d x d y$

$\text { Identify the surface with equation } \phi = \frac { \pi } { 6 }$

Use spherical coordinates to evaluate $\iiint _ { B } \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d V$ , where $B$ is the ball $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 7$ . Select the correct answer.

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 . Select the correct answer.

Find the area of the surface $S$ where $S$ is the part of the plane $z = 3 x ^ { 2 } + y$ that lies above the triangular region with vertices $( 0,0 ) , ( 2,0 )$ , and $( 2,2 )$ .

Find the mass and the center of mass of the lamina occupying the region $R$ , where $R$ is the triangular region with vertices $( 0,0 ) , ( 3,8 )$ , and $( 6,0 )$ , and having the mass density $\rho ( x , y ) = x$ .

Use the Midpoint Rule with four squares of equal size to estimate the double integral. $\iint _ { R } \cos \left( x ^ { 4 } + y ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 0.4,0 \leq y \leq 0.4 \}$

Evaluate the double integral by first identifying it as the volume of a solid. $\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 4 \leq x \leq 7,2 \leq y \leq 6 \}$ Select the correct answer.

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

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

Calculate the double integral. Select the correct answer. $\iint _ { R } \left( 15 x ^ { 2 } y ^ { 3 } - 25 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}$

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 $x y = 2 , x y = 4 ; x = \frac { u } { v } , y = v$ . Select the correct answer.

Find the Jacobian of the transformation. $x = 5 \alpha \sin \beta , y = 4 \alpha \cos \beta$

Evaluate the integral $\iiint _ { B } f ( x , y , z ) d V$ where $f ( x , y , z ) = x y ^ { 2 } + y z ^ { 2 }$ and $B = \{ ( x , y , z ) \mid 0 \leq x \leq 2 , - 1 \leq y \leq 1,0 \leq z \leq 3 \}$ with respect to $x , y$ , and $z$ , in that order.

Use the transformation $x = \sqrt { 2 } u - \sqrt { \frac { 2 } { 3 } } v , y = \sqrt { 2 } u + \sqrt { \frac { 2 } { 3 } } v$ 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 $x ^ { 2 } - x y + y ^ { 2 } = 2$ .

Calculate the iterated integral. Round your answer to two decimal places. $\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { 3 } \sqrt { 25 x + 25 y } d x d y$

Calculate the iterated integral. $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } 5 \cos \left( y ^ { 2 } \right) d x d y$

Evaluate the integral by reversing the order of integration. $\int _ { 0 } ^ { 1 } \int _ { 2 n } ^ { 2 } e ^ { x ^ { 2 } } d x d y$

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 $x ^ { 2 } + y ^ { 2 } = 36$ .

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Vector Calculus

Second-Order Differential Equations

Filters

Exam 15: Multiple Integrals

Evaluate the iterated integral. ∫13∫y310xydxdy\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 10 x y d x d y∫13​∫y3​10xydxdy

Identify the surface with equation ϕ=π6\text { Identify the surface with equation } \phi = \frac { \pi } { 6 } Identify the surface with equation ϕ=6π​

Use spherical coordinates to evaluate ∭Bx2+y2+z2dV\iiint _ { B } \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d V∭B​x2+y2+z2​dV , where BBB is the ball x2+y2+z2≤7x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 7x2+y2+z2≤7 . Select the correct answer.

Find the mass of the solid SSS bounded by the paraboloid z=6x2+6y2z = 6 x ^ { 2 } + 6 y ^ { 2 }z=6x2+6y2 and the plane z=5z = 5z=5 if SSS has constant density 3 . Select the correct answer.

Find the area of the surface SSS where SSS is the part of the plane z=3x2+yz = 3 x ^ { 2 } + yz=3x2+y that lies above the triangular region with vertices (0,0),(2,0)( 0,0 ) , ( 2,0 )(0,0),(2,0) , and (2,2)( 2,2 )(2,2) .

Find the mass and the center of mass of the lamina occupying the region RRR , where RRR is the triangular region with vertices (0,0),(3,8)( 0,0 ) , ( 3,8 )(0,0),(3,8) , and (6,0)( 6,0 )(6,0) , and having the mass density ρ(x,y)=x\rho ( x , y ) = xρ(x,y)=x .

Evaluate the double integral by first identifying it as the volume of a solid. ∬R(15−2x)dA,R={(x,y)∣4≤x≤7,2≤y≤6}\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 4 \leq x \leq 7,2 \leq y \leq 6 \}∬R​(15−2x)dA,R={(x,y)∣4≤x≤7,2≤y≤6} Select the correct answer.

Use polar coordinates to find the volume of the solid inside the cylinder x2+y2=16x ^ { 2 } + y ^ { 2 } = 16x2+y2=16 and the ellipsoid 2x2+2y2+z2=642 x ^ { 2 } + 2 y ^ { 2 } + z ^ { 2 } = 642x2+2y2+z2=64 .

Find the mass and the center of mass of the lamina occupying the region RRR , where RRR is the triangular region with vertices (0,0),(2,5)( 0,0 ) , ( 2,5 )(0,0),(2,5) , and (4,0)( 4,0 )(4,0) , and having the mass density ρ(x,y)=x\rho ( x , y ) = xρ(x,y)=x . Select the correct answer.

Calculate the double integral. Select the correct answer. ∬R(15x2y3−25x4)dA,R={(x,y)∣0≤x≤1,0≤y≤4}\iint _ { R } \left( 15 x ^ { 2 } y ^ { 3 } - 25 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}∬R​(15x2y3−25x4)dA,R={(x,y)∣0≤x≤1,0≤y≤4}

Find the Jacobian of the transformation. x=5αsin⁡β,y=4αcos⁡βx = 5 \alpha \sin \beta , y = 4 \alpha \cos \betax=5αsinβ,y=4αcosβ

Calculate the iterated integral. Round your answer to two decimal places. ∫06∫0325x+25ydxdy\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { 3 } \sqrt { 25 x + 25 y } d x d y∫06​∫03​25x+25y​dxdy

Calculate the iterated integral. ∫01∫0y5cos⁡(y2)dxdy\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } 5 \cos \left( y ^ { 2 } \right) d x d y∫01​∫0y​5cos(y2)dxdy

Evaluate the integral by reversing the order of integration. ∫01∫2n2ex2dxdy\int _ { 0 } ^ { 1 } \int _ { 2 n } ^ { 2 } e ^ { x ^ { 2 } } d x d y∫01​∫2n2​ex2dxdy

Find the area of the part of hyperbolic paraboloid z=y2−x2z = y ^ { 2 } - x ^ { 2 }z=y2−x2 that lies between the cylinders x2+y2=1x ^ { 2 } + y ^ { 2 } = 1x2+y2=1 and x2+y2=36x ^ { 2 } + y ^ { 2 } = 36x2+y2=36 .

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Vector Calculus

Second-Order Differential Equations

Filters

Evaluate the iterated integral. $\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 10 x y d x d y$

$\text { Identify the surface with equation } \phi = \frac { \pi } { 6 }$

Use spherical coordinates to evaluate $\iiint _ { B } \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d V$ , where $B$ is the ball $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 7$ . Select the correct answer.

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 . Select the correct answer.

Find the area of the surface $S$ where $S$ is the part of the plane $z = 3 x ^ { 2 } + y$ that lies above the triangular region with vertices $( 0,0 ) , ( 2,0 )$ , and $( 2,2 )$ .

Find the mass and the center of mass of the lamina occupying the region $R$ , where $R$ is the triangular region with vertices $( 0,0 ) , ( 3,8 )$ , and $( 6,0 )$ , and having the mass density $\rho ( x , y ) = x$ .

Evaluate the double integral by first identifying it as the volume of a solid. $\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 4 \leq x \leq 7,2 \leq y \leq 6 \}$ Select the correct answer.

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

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

Calculate the double integral. Select the correct answer. $\iint _ { R } \left( 15 x ^ { 2 } y ^ { 3 } - 25 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}$

Find the Jacobian of the transformation. $x = 5 \alpha \sin \beta , y = 4 \alpha \cos \beta$

Calculate the iterated integral. Round your answer to two decimal places. $\int _ { 0 } ^ { 6 } \int _ { 0 } ^ { 3 } \sqrt { 25 x + 25 y } d x d y$

Calculate the iterated integral. $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } 5 \cos \left( y ^ { 2 } \right) d x d y$

Evaluate the integral by reversing the order of integration. $\int _ { 0 } ^ { 1 } \int _ { 2 n } ^ { 2 } e ^ { x ^ { 2 } } d x d y$

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 $x ^ { 2 } + y ^ { 2 } = 36$ .