Question 1

$$\text { Find the volume under } z = 4 x ^ { 5 } + 4 y ^ { 5 } \text { and above the region bounded by } y = x ^ { 2 } \text { and } x = y ^ { 2 } \text {. }$$

Accepted Answer

The answer of \[\text { Find the volume under }...

Question 2

Calculate the double integral. Round your answer to two decimal places.
$$\iint _ { R } \frac { 4 + x ^ { 2 } } { 1 + y ^ { 2 } } d A , R = \{ ( x , y ) \mid 0 \leq x \leq 6,0 \leq y \leq 4 \}$$

Accepted Answer

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

Question 3

Find the Jacobian of the transformation.
$$x = 7 \alpha \sin \beta , y = 6 \alpha \cos \beta$$

Accepted Answer

The answer of Find the Jacobian of the transformation.
\[x =...

Question 4

Find the mass of the lamina that occupies the region $D$ and has the given density function. Round your answer to two decimal places.
$$R = \{ ( x , y ) \mid 1 \leq x \leq 3,1 \leq y \leq 4 \} \rho ( x , y ) = 5 y ^ { 2 }$$

Accepted Answer

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

Question 5

Find the mass and the center of mass of the lamina occupying the region $R$, where $R$ is the region bounded by the graphs of $y = \sin 6 x , y = 0 , x = 0$, and $x = \frac { \pi } { 6 }$, and having the mass density $\rho ( x , y ) = 4 y$.

Accepted Answer

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

Question 6

Evaluate the double integral $\iint _ { R } 3 x ^ { 2 } d A$, where $R$ is the region bounded by the graphs of $y = ( x - 1 ) ^ { 2 }$ and $y = - x + 3$

Accepted Answer

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

Question 7

Evaluate the double integral by first identifying it as the volume of a solid.
$$\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 3 \leq x \leq 7,3 \leq y \leq 5 \}$$

Accepted Answer

A)  40 
B)  $- 160$ 
C)  $- 60$ 
D)  140 
E)  240 
A)  40 
B)  $- 160$ 
C)  $- 60$ 
D)  140 
E)  240

Question 8

Find the center of mass of the lamina that occupies the region $D$ and has the given density function, if $D$ is bounded by the parabola $y = 1 - x ^ { 2 }$ and the $x$-axis.
$$\rho ( x , y ) = 4 y$$

Accepted Answer

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

Question 9

Use polar coordinates to find the volume of the solid under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and above the disk $x ^ { 2 } + y ^ { 2 } \leq 25$.

Accepted Answer

A)  $2,500 \pi$ 
B)  $625 \pi$ 
C)  $312.5 \pi$ 
D)  $1,250 \pi$ 
E)  $208.3 \pi$ 
A)  $2,500 \pi$ 
B)  $625 \pi$ 
C)  $312.5 \pi$ 
D)  $1,250 \pi$ 
E)  $208.3 \pi$

Question 10

$$\text { Use spherical coordinates to evaluate } \iiint _ { B } \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d V \text {, where } B \text { is the ball } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 6 \text {. }$$

Accepted Answer

The answer of \[\text { Use spherical coordinates to evaluate...

Question 11

Calculate the double integral.
$$\iint _ { R } \left( 9 x ^ { 2 } y ^ { 3 } - 15 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}$$

Accepted Answer

A)  250 
B)  230 
C)  190 
D)  180 
E)  210 
A)  250 
B)  230 
C)  190 
D)  180 
E)  210

Question 12

Calculate the iterated integral.
$$\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { 1 } 5 y e ^ { x y } d x d y$$

Accepted Answer

A)  $\frac { \left( 5 e ^ { 2 } + 10 e + 5 \right) } { e }$ 
B)  $\frac { \left( 5 e ^ { 2 } + 10 e - 5 \right) } { e }$ 
C)  $5 e ^ { 2 } + 10 e$ 
D)  $5 e ^ { 2 } + 10 e + 5$ 
E)  $\frac { \left( 5 e ^ { 2 } - 10 e - 5 \right) } { e }$ 
A)  $\frac { \left( 5 e ^ { 2 } + 10 e + 5 \right) } { e }$ 
B)  $\frac { \left( 5 e ^ { 2 } + 10 e - 5 \right) } { e }$ 
C)  $5 e ^ { 2 } + 10 e$ 
D)  $5 e ^ { 2 } + 10 e + 5$ 
E)  $\frac { \left( 5 e ^ { 2 } - 10 e - 5 \right) } { e }$

Question 13

Find the volume of the solid bounded by the surface $z = 5 + ( x - 4 ) ^ { 2 } + 2 y$ and the planes $x = 3 , y = 4$ and coordinate planes.

Accepted Answer

A)  $V = 768$ 
B)  $V = 256$ 
C)  $V = 384$ 
D)  $V = 192$ 
E)  $V = 576$ 
A)  $V = 768$ 
B)  $V = 256$ 
C)  $V = 384$ 
D)  $V = 192$ 
E)  $V = 576$

Question 14

Evaluate the triple integral. Round your answer to one decimal place.
$$\iiint _ { B } 4 x y d V$$
$E$ lies under the plane $z = 5 + x + y$ and above the region in the $x y$-plane bounded by the curves $y = \sqrt { x } , y = 0$, and $x = 4$.

Accepted Answer

The answer of Evaluate the triple integral. Round your answer...

Question 15

Find the area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36 z$ that lies inside the paraboloid $z = x ^ { 2 } + y ^ { 2 }$.
Select the correct answer.

Accepted Answer

A)  $6 \pi$ 
B)  $36 \pi$ 
C)  $12.5 \pi$ 
D)  $16 \pi$ 
E)  $10.5 \pi$ 
A)  $6 \pi$ 
B)  $36 \pi$ 
C)  $12.5 \pi$ 
D)  $16 \pi$ 
E)  $10.5 \pi$

Question 16

Use spherical coordinates to find the moment of inertia of the solid homogeneous hemisphere of radius 5 and density 1 about a diameter of its base. Select the correct answer.

Accepted Answer

A)  $205.13$ 
B)  $2616.67$ 
C)  $195.22$ 
D)  $213.5$ 
E)  $198.08$ 
A)  $205.13$ 
B)  $2616.67$ 
C)  $195.22$ 
D)  $213.5$ 
E)  $198.08$

Question 17

Find the volume of the solid bounded by the surface $z = 5 + ( x - 4 ) ^ { 2 } + 2 y$ and the planes $x = 3 , y = 4$ and coordinate planes.

Accepted Answer

The answer of Find the volume of the solid bounded...

Question 18

Determine whether to use polar coordinates or rectangular coordinates to evaluate the integral $\iint _ { R } f ( x , y ) d A$, where $f$ is a continuous function. Then write an expression for the (iterated) integral.

Accepted Answer

The answer of Determine whether to use polar coordinates or...

Question 19

Use cylindrical coordinates to evaluate $\int _ { - 2 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - x ^ { 2 } } } \int _ { 0 } ^ { \sqrt { 16 - x ^ { 2 } - y ^ { 2 } } } z d z d y d x$.
Select the correct answer.

Accepted Answer

A)  $12 \pi$ 
B)  $112 \pi$ 
C)  $\frac { 3 } { 2 } \pi$ 
D)  $14 \pi$ 
A)  $12 \pi$ 
B)  $112 \pi$ 
C)  $\frac { 3 } { 2 } \pi$ 
D)  $14 \pi$

Question 20

Determine whether to use polar coordinates or rectangular coordinates to evaluate the integral $\iint _ { R } f ( x , y ) d A$, where $f$ is a continuous function. Then write an expression for the (iterated) integral.

Accepted Answer

The answer of Determine whether to use polar coordinates or...

$\text { Find the volume under } z = 4 x ^ { 5 } + 4 y ^ { 5 } \text { and above the region bounded by } y = x ^ { 2 } \text { and } x = y ^ { 2 } \text {. }$

Calculate the double integral. Round your answer to two decimal places. $\iint _ { R } \frac { 4 + x ^ { 2 } } { 1 + y ^ { 2 } } d A , R = \{ ( x , y ) \mid 0 \leq x \leq 6,0 \leq y \leq 4 \}$

Find the Jacobian of the transformation. $x = 7 \alpha \sin \beta , y = 6 \alpha \cos \beta$

Find the mass of the lamina that occupies the region $D$ and has the given density function. Round your answer to two decimal places. $R = \{ ( x , y ) \mid 1 \leq x \leq 3,1 \leq y \leq 4 \} \rho ( x , y ) = 5 y ^ { 2 }$

Find the mass and the center of mass of the lamina occupying the region $R$ , where $R$ is the region bounded by the graphs of $y = \sin 6 x , y = 0 , x = 0$ , and $x = \frac { \pi } { 6 }$ , and having the mass density $\rho ( x , y ) = 4 y$ .

Evaluate the double integral $\iint _ { R } 3 x ^ { 2 } d A$ , where $R$ is the region bounded by the graphs of $y = ( x - 1 ) ^ { 2 }$ and $y = - x + 3$

Evaluate the double integral by first identifying it as the volume of a solid. $\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 3 \leq x \leq 7,3 \leq y \leq 5 \}$

Find the center of mass of the lamina that occupies the region $D$ and has the given density function, if $D$ is bounded by the parabola $y = 1 - x ^ { 2 }$ and the $x$ -axis. $\rho ( x , y ) = 4 y$

Use polar coordinates to find the volume of the solid under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and above the disk $x ^ { 2 } + y ^ { 2 } \leq 25$ .

$\text { Use spherical coordinates to evaluate } \iiint _ { B } \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d V \text {, where } B \text { is the ball } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 6 \text {. }$

Calculate the double integral. $\iint _ { R } \left( 9 x ^ { 2 } y ^ { 3 } - 15 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}$

Calculate the iterated integral. $\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { 1 } 5 y e ^ { x y } d x d y$

Find the volume of the solid bounded by the surface $z = 5 + ( x - 4 ) ^ { 2 } + 2 y$ and the planes $x = 3 , y = 4$ and coordinate planes.

Evaluate the triple integral. Round your answer to one decimal place. $\iiint _ { B } 4 x y d V$ $E$ lies under the plane $z = 5 + x + y$ and above the region in the $x y$ -plane bounded by the curves $y = \sqrt { x } , y = 0$ , and $x = 4$ .

Find the area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36 z$ that lies inside the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ . Select the correct answer.

Use spherical coordinates to find the moment of inertia of the solid homogeneous hemisphere of radius 5 and density 1 about a diameter of its base. Select the correct answer.

Find the volume of the solid bounded by the surface $z = 5 + ( x - 4 ) ^ { 2 } + 2 y$ and the planes $x = 3 , y = 4$ and coordinate planes.

Use cylindrical coordinates to evaluate $\int _ { - 2 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - x ^ { 2 } } } \int _ { 0 } ^ { \sqrt { 16 - x ^ { 2 } - y ^ { 2 } } } z d z d y d x$ . Select the correct answer.

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Exam 15: Multiple Integrals

Calculate the double integral. Round your answer to two decimal places. ∬R4+x21+y2dA,R={(x,y)∣0≤x≤6,0≤y≤4}\iint _ { R } \frac { 4 + x ^ { 2 } } { 1 + y ^ { 2 } } d A , R = \{ ( x , y ) \mid 0 \leq x \leq 6,0 \leq y \leq 4 \}∬R​1+y24+x2​dA,R={(x,y)∣0≤x≤6,0≤y≤4}

Find the Jacobian of the transformation. x=7αsin⁡β,y=6αcos⁡βx = 7 \alpha \sin \beta , y = 6 \alpha \cos \betax=7αsinβ,y=6αcosβ

Evaluate the double integral ∬R3x2dA\iint _ { R } 3 x ^ { 2 } d A∬R​3x2dA , where RRR is the region bounded by the graphs of y=(x−1)2y = ( x - 1 ) ^ { 2 }y=(x−1)2 and y=−x+3y = - x + 3y=−x+3

Evaluate the double integral by first identifying it as the volume of a solid. ∬R(15−2x)dA,R={(x,y)∣3≤x≤7,3≤y≤5}\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 3 \leq x \leq 7,3 \leq y \leq 5 \}∬R​(15−2x)dA,R={(x,y)∣3≤x≤7,3≤y≤5}

Find the center of mass of the lamina that occupies the region DDD and has the given density function, if DDD is bounded by the parabola y=1−x2y = 1 - x ^ { 2 }y=1−x2 and the xxx -axis. ρ(x,y)=4y\rho ( x , y ) = 4 yρ(x,y)=4y

Use polar coordinates to find the volume of the solid under the paraboloid z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 and above the disk x2+y2≤25x ^ { 2 } + y ^ { 2 } \leq 25x2+y2≤25 .

Calculate the double integral. ∬R(9x2y3−15x4)dA,R={(x,y)∣0≤x≤1,0≤y≤4}\iint _ { R } \left( 9 x ^ { 2 } y ^ { 3 } - 15 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}∬R​(9x2y3−15x4)dA,R={(x,y)∣0≤x≤1,0≤y≤4}

Calculate the iterated integral. ∫−11∫015yexydxdy\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { 1 } 5 y e ^ { x y } d x d y∫−11​∫01​5yexydxdy

Find the volume of the solid bounded by the surface z=5+(x−4)2+2yz = 5 + ( x - 4 ) ^ { 2 } + 2 yz=5+(x−4)2+2y and the planes x=3,y=4x = 3 , y = 4x=3,y=4 and coordinate planes.

Evaluate the triple integral. Round your answer to one decimal place. ∭B4xydV\iiint _ { B } 4 x y d V∭B​4xydV EEE lies under the plane z=5+x+yz = 5 + x + yz=5+x+y and above the region in the xyx yxy -plane bounded by the curves y=x,y=0y = \sqrt { x } , y = 0y=x​,y=0 , and x=4x = 4x=4 .

Find the area of the part of the sphere x2+y2+z2=36zx ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36 zx2+y2+z2=36z that lies inside the paraboloid z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 . Select the correct answer.

Use spherical coordinates to find the moment of inertia of the solid homogeneous hemisphere of radius 5 and density 1 about a diameter of its base. Select the correct answer.

Find the volume of the solid bounded by the surface z=5+(x−4)2+2yz = 5 + ( x - 4 ) ^ { 2 } + 2 yz=5+(x−4)2+2y and the planes x=3,y=4x = 3 , y = 4x=3,y=4 and coordinate planes.

Determine whether to use polar coordinates or rectangular coordinates to evaluate the integral ∬Rf(x,y)dA\iint _ { R } f ( x , y ) d A∬R​f(x,y)dA , where fff is a continuous function. Then write an expression for the (iterated) integral.

Determine whether to use polar coordinates or rectangular coordinates to evaluate the integral ∬Rf(x,y)dA\iint _ { R } f ( x , y ) d A∬R​f(x,y)dA , where fff is a continuous function. Then write an expression for the (iterated) integral.

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Vector Functions

Partial Derivatives

Vector Calculus

Second-Order Differential Equations

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Calculate the double integral. Round your answer to two decimal places. $\iint _ { R } \frac { 4 + x ^ { 2 } } { 1 + y ^ { 2 } } d A , R = \{ ( x , y ) \mid 0 \leq x \leq 6,0 \leq y \leq 4 \}$

Find the Jacobian of the transformation. $x = 7 \alpha \sin \beta , y = 6 \alpha \cos \beta$

Evaluate the double integral $\iint _ { R } 3 x ^ { 2 } d A$ , where $R$ is the region bounded by the graphs of $y = ( x - 1 ) ^ { 2 }$ and $y = - x + 3$

Evaluate the double integral by first identifying it as the volume of a solid. $\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 3 \leq x \leq 7,3 \leq y \leq 5 \}$

Find the center of mass of the lamina that occupies the region $D$ and has the given density function, if $D$ is bounded by the parabola $y = 1 - x ^ { 2 }$ and the $x$ -axis. $\rho ( x , y ) = 4 y$

Use polar coordinates to find the volume of the solid under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and above the disk $x ^ { 2 } + y ^ { 2 } \leq 25$ .

Calculate the double integral. $\iint _ { R } \left( 9 x ^ { 2 } y ^ { 3 } - 15 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}$

Calculate the iterated integral. $\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { 1 } 5 y e ^ { x y } d x d y$

Find the volume of the solid bounded by the surface $z = 5 + ( x - 4 ) ^ { 2 } + 2 y$ and the planes $x = 3 , y = 4$ and coordinate planes.

Evaluate the triple integral. Round your answer to one decimal place. $\iiint _ { B } 4 x y d V$ $E$ lies under the plane $z = 5 + x + y$ and above the region in the $x y$ -plane bounded by the curves $y = \sqrt { x } , y = 0$ , and $x = 4$ .

Find the area of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36 z$ that lies inside the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ . Select the correct answer.

Find the volume of the solid bounded by the surface $z = 5 + ( x - 4 ) ^ { 2 } + 2 y$ and the planes $x = 3 , y = 4$ and coordinate planes.