Exam 15: Multiple Integrals

Evaluate the integral by changing to polar coordinates. $\iint _ { D } e ^ { - x ^ { 2 } - y ^ { 2 } } d A$ $D$ is the region bounded by the semicircle $x = \sqrt { 9 - y ^ { 2 } }$ and the $y$ -axis.

(Short Answer)

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Question 61

Use a triple integral to find the volume of the solid bounded by $x = y ^ { 2 }$ and the planes $z = 0$ and $x + z = 3$ . Select the correct answer.

(Multiple Choice)

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Question 62

Find the center of mass of the lamina of the region shown if the density of the circular lamina is four times that of the rectangular lamina.

(Short Answer)

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Question 63

Use a computer algebra system to find the moment of inertia $I _ { 0 }$ of the lamina that occupies the region $D$ and has the density function $\rho ( x , y ) = 3 x y$ , if $D = \{ ( x , y ) \mid 0 \leq x \leq \pi , 0 \leq y \leq \sin ( x ) \}$ .

(Multiple Choice)

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Question 64

The sketch of the solid is given below. Given $a = 5$ , write the inequalities that describe it. Select the correct answer.

(Multiple Choice)

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Question 65

Find the center of mass of the system comprising masses $m _ { k }$ located at the points $P _ { k }$ in a coordinate plane. Assume that mass is measured in grams and distance is measured in centimeters. =4,=3,=1 (3,-3),(5,-1),(2,-5)

(Short Answer)

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Question 66

Find the area of the surface. Round your answer to three decimal places. $z = \frac { 2 } { 3 } \left( x ^ { 3 / 2 } + y ^ { 3 / 2 } \right) , 0 \leq x \leq 3,0 \leq y \leq 3$

(Short Answer)

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Question 67

Use spherical coordinate to find the volume above the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ and inside sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2 a z$ .

(Short Answer)

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Question 68

Calculate the iterated integral. $\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 4 y \sin x d z d y d x$

(Multiple Choice)

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Question 69

Calculate the iterated integral. $\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 4 y \sin x d z d y d x$

(Short Answer)

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Question 70

$\text { Sketch the solid whose volume is given by the iterated integral } \int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 2 - x } \int _ { 0 } ^ { 2 - x - y } f ( x , y , z ) d z d y d x \text {. }$

(Essay)

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Question 71

Calculate the double integral. $\iint _ { R } \left( 6 x ^ { 2 } y ^ { 3 } - 10 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}$

(Multiple Choice)

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Question 72

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. Select the correct answer.

(Multiple Choice)

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Question 73

Use spherical coordinate to find the volume above the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ and inside sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2 a z$ .

(Short Answer)

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Question 74

Evaluate the integral $\iint _ { R } \frac { x ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } d A$ , where $R$ is the annular region bounded by the circles $x ^ { 2 } + y ^ { 2 } = 9$ and $x ^ { 2 } + y ^ { 2 } = 16$ , by changing to polar coordinates.

(Short Answer)

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Question 75

$\text { The sketch of the solid is given below. Given } a = 4 \text {, write the inequalities that describe it. }$ $\text { The sketch of the solid is given below. Given } a = 4 \text {, write the inequalities that describe it. }$

(Short Answer)

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Question 76

Use cylindrical coordinates to evaluate the triple integral $\iiint _ { E } y d V$ where $E$ is the solid that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 3$ and $x ^ { 2 } + y ^ { 2 } = 7$ above the $x y$ -plane and below the plane $z = x + 4$ .

(Multiple Choice)

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Question 77

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.

(Multiple Choice)

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Question 78

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

(Short Answer)

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Question 79

Find the mass and the center of mass of the lamina occupying the region $R$ , where $R$ is the triangular region with vertices $( 0,0 ) , ( 5,2 )$ , and $( 10,0 )$ , and having the mass density $\rho ( x , y ) = x$ .

(Multiple Choice)

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Question 80

Evaluate the integral by changing to polar coordinates. $\iint _ { D } e ^ { - x ^ { 2 } - y ^ { 2 } } d A$ $D$ is the region bounded by the semicircle $x = \sqrt { 9 - y ^ { 2 } }$ and the $y$ -axis.

Use a triple integral to find the volume of the solid bounded by $x = y ^ { 2 }$ and the planes $z = 0$ and $x + z = 3$ . Select the correct answer.

Find the center of mass of the lamina of the region shown if the density of the circular lamina is four times that of the rectangular lamina.

Use a computer algebra system to find the moment of inertia $I _ { 0 }$ of the lamina that occupies the region $D$ and has the density function $\rho ( x , y ) = 3 x y$ , if $D = \{ ( x , y ) \mid 0 \leq x \leq \pi , 0 \leq y \leq \sin ( x ) \}$ .

The sketch of the solid is given below. Given $a = 5$ , write the inequalities that describe it. Select the correct answer.

Find the center of mass of the system comprising masses $m _ { k }$ located at the points $P _ { k }$ in a coordinate plane. Assume that mass is measured in grams and distance is measured in centimeters. =4,=3,=1 (3,-3),(5,-1),(2,-5)

Find the area of the surface. Round your answer to three decimal places. $z = \frac { 2 } { 3 } \left( x ^ { 3 / 2 } + y ^ { 3 / 2 } \right) , 0 \leq x \leq 3,0 \leq y \leq 3$

Use spherical coordinate to find the volume above the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ and inside sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2 a z$ .

Calculate the iterated integral. $\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 4 y \sin x d z d y d x$

Calculate the iterated integral. $\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 4 y \sin x d z d y d x$

$\text { Sketch the solid whose volume is given by the iterated integral } \int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 2 - x } \int _ { 0 } ^ { 2 - x - y } f ( x , y , z ) d z d y d x \text {. }$

Calculate the double integral. $\iint _ { R } \left( 6 x ^ { 2 } y ^ { 3 } - 10 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}$

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. Select the correct answer.

Use spherical coordinate to find the volume above the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ and inside sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2 a z$ .

Evaluate the integral $\iint _ { R } \frac { x ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } d A$ , where $R$ is the annular region bounded by the circles $x ^ { 2 } + y ^ { 2 } = 9$ and $x ^ { 2 } + y ^ { 2 } = 16$ , by changing to polar coordinates.

$\text { The sketch of the solid is given below. Given } a = 4 \text {, write the inequalities that describe it. }$ $\text { The sketch of the solid is given below. Given } a = 4 \text {, write the inequalities that describe it. }$

Use cylindrical coordinates to evaluate the triple integral $\iiint _ { E } y d V$ where $E$ is the solid that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 3$ and $x ^ { 2 } + y ^ { 2 } = 7$ above the $x y$ -plane and below the plane $z = x + 4$ .

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.

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

Find the mass and the center of mass of the lamina occupying the region $R$ , where $R$ is the triangular region with vertices $( 0,0 ) , ( 5,2 )$ , and $( 10,0 )$ , and having the mass density $\rho ( x , y ) = x$ .

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Exam 15: Multiple Integrals

Evaluate the integral by changing to polar coordinates. ∬De−x2−y2dA\iint _ { D } e ^ { - x ^ { 2 } - y ^ { 2 } } d A∬D​e−x2−y2dA DDD is the region bounded by the semicircle x=9−y2x = \sqrt { 9 - y ^ { 2 } }x=9−y2​ and the yyy -axis.

Use a triple integral to find the volume of the solid bounded by x=y2x = y ^ { 2 }x=y2 and the planes z=0z = 0z=0 and x+z=3x + z = 3x+z=3 . Select the correct answer.

Find the center of mass of the lamina of the region shown if the density of the circular lamina is four times that of the rectangular lamina.

The sketch of the solid is given below. Given a=5a = 5a=5 , write the inequalities that describe it. Select the correct answer.

Find the center of mass of the system comprising masses mkm _ { k }mk​ located at the points PkP _ { k }Pk​ in a coordinate plane. Assume that mass is measured in grams and distance is measured in centimeters. =4,=3,=1 (3,-3),(5,-1),(2,-5)

Find the area of the surface. Round your answer to three decimal places. z=23(x3/2+y3/2),0≤x≤3,0≤y≤3z = \frac { 2 } { 3 } \left( x ^ { 3 / 2 } + y ^ { 3 / 2 } \right) , 0 \leq x \leq 3,0 \leq y \leq 3z=32​(x3/2+y3/2),0≤x≤3,0≤y≤3

Use spherical coordinate to find the volume above the cone z2=x2+y2z ^ { 2 } = x ^ { 2 } + y ^ { 2 }z2=x2+y2 and inside sphere x2+y2+z2=2azx ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2 a zx2+y2+z2=2az .

Calculate the iterated integral. ∫0x∫01∫01−y24ysin⁡xdzdydx\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 4 y \sin x d z d y d x∫0x​∫01​∫01−y2​​4ysinxdzdydx

Calculate the iterated integral. ∫0x∫01∫01−y24ysin⁡xdzdydx\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 4 y \sin x d z d y d x∫0x​∫01​∫01−y2​​4ysinxdzdydx

Calculate the double integral. ∬R(6x2y3−10x4)dA,R={(x,y)∣0≤x≤1,0≤y≤4}\iint _ { R } \left( 6 x ^ { 2 } y ^ { 3 } - 10 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}∬R​(6x2y3−10x4)dA,R={(x,y)∣0≤x≤1,0≤y≤4}

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. Select the correct answer.

Use spherical coordinate to find the volume above the cone z2=x2+y2z ^ { 2 } = x ^ { 2 } + y ^ { 2 }z2=x2+y2 and inside sphere x2+y2+z2=2azx ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2 a zx2+y2+z2=2az .

Evaluate the integral ∬Rx2x2+y2dA\iint _ { R } \frac { x ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } d A∬R​x2+y2x2​dA , where RRR is the annular region bounded by the circles x2+y2=9x ^ { 2 } + y ^ { 2 } = 9x2+y2=9 and x2+y2=16x ^ { 2 } + y ^ { 2 } = 16x2+y2=16 , by changing to polar coordinates.

The sketch of the solid is given below. Given a=4, write the inequalities that describe it. \text { The sketch of the solid is given below. Given } a = 4 \text {, write the inequalities that describe it. } The sketch of the solid is given below. Given a=4, write the inequalities that describe it.

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

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),(5,2)( 0,0 ) , ( 5,2 )(0,0),(5,2) , and (10,0)( 10,0 )(10,0) , and having the mass density ρ(x,y)=x\rho ( x , y ) = xρ(x,y)=x .

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Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

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Vectors and the Geometry of Space

Vector Functions

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Evaluate the integral by changing to polar coordinates. $\iint _ { D } e ^ { - x ^ { 2 } - y ^ { 2 } } d A$ $D$ is the region bounded by the semicircle $x = \sqrt { 9 - y ^ { 2 } }$ and the $y$ -axis.

Use a triple integral to find the volume of the solid bounded by $x = y ^ { 2 }$ and the planes $z = 0$ and $x + z = 3$ . Select the correct answer.

The sketch of the solid is given below. Given $a = 5$ , write the inequalities that describe it. Select the correct answer.

Find the center of mass of the system comprising masses $m _ { k }$ located at the points $P _ { k }$ in a coordinate plane. Assume that mass is measured in grams and distance is measured in centimeters. =4,=3,=1 (3,-3),(5,-1),(2,-5)

Find the area of the surface. Round your answer to three decimal places. $z = \frac { 2 } { 3 } \left( x ^ { 3 / 2 } + y ^ { 3 / 2 } \right) , 0 \leq x \leq 3,0 \leq y \leq 3$

Use spherical coordinate to find the volume above the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ and inside sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2 a z$ .

Calculate the iterated integral. $\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 4 y \sin x d z d y d x$

Calculate the iterated integral. $\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 4 y \sin x d z d y d x$

Calculate the double integral. $\iint _ { R } \left( 6 x ^ { 2 } y ^ { 3 } - 10 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}$

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. Select the correct answer.

Use spherical coordinate to find the volume above the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ and inside sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2 a z$ .

Evaluate the integral $\iint _ { R } \frac { x ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } d A$ , where $R$ is the annular region bounded by the circles $x ^ { 2 } + y ^ { 2 } = 9$ and $x ^ { 2 } + y ^ { 2 } = 16$ , by changing to polar coordinates.

$\text { The sketch of the solid is given below. Given } a = 4 \text {, write the inequalities that describe it. }$ $\text { The sketch of the solid is given below. Given } a = 4 \text {, write the inequalities that describe it. }$

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

Find the mass and the center of mass of the lamina occupying the region $R$ , where $R$ is the triangular region with vertices $( 0,0 ) , ( 5,2 )$ , and $( 10,0 )$ , and having the mass density $\rho ( x , y ) = x$ .