Question 1

Find the Jacobian of the transformation $x = u ^ { 2 }$ , $y = v ^ { 3 }$ .

Accepted Answer

A)  $u v ^ { 2 }$ 
B)  $2 u v ^ { 2 }$ 
C)  $3 u v ^ { 2 }$ 
D) $6 u v ^ { 2 }$ 
E)  $- u v ^ { 2 }$ 
F) $- 2 u v ^ { 2 }$

G) $- 3 u v ^ { 2 }$ 
 
H) $- 6 u v ^ { 2 }$ 
A)  $u v ^ { 2 }$ 
B)  $2 u v ^ { 2 }$ 
C)  $3 u v ^ { 2 }$ 
D) $6 u v ^ { 2 }$ 
E)  $- u v ^ { 2 }$ 
F) $- 2 u v ^ { 2 }$

G) $- 3 u v ^ { 2 }$ 
 
H) $- 6 u v ^ { 2 }$

Question 2

Find $\int _ { 0 } ^ { 2 } f ( x , y ) d y$ and $\int _ { 0 } ^ { 1 } f ( x , y ) d x$ for $f ( x , y ) = 2 x y - 3 x ^ { 2 }$ .

Accepted Answer

The answer of Find \(\int _ { 0 } ^...

Question 3

A solid is bounded above by the paraboloid $z = 1 - x ^ { 2 } - y ^ { 2 }$ and below by the xy-plane. Compute the volume of this solid using polar coordinates.

Accepted Answer

The answer of A solid is bounded above by the...

Question 4

Use a double integral to find the area of the region enclosed by one loop of the curve $r = \cos 2 \theta$ .

Accepted Answer

A) 1 
B)  $\frac { \pi } { 4 }$ 
C)  $\frac { \pi } { 16 }$ 
D)  $\frac { 46 \pi } { 27 }$ 
E)  $\frac { 5 } { 2 }$
F) $\frac { 1 } { 2 }$ 
G) $\frac { \pi } { 8 }$ 
H) $\frac { 46 } { 27 }$ 
A) 1 
B)  $\frac { \pi } { 4 }$ 
C)  $\frac { \pi } { 16 }$ 
D)  $\frac { 46 \pi } { 27 }$ 
E)  $\frac { 5 } { 2 }$
F) $\frac { 1 } { 2 }$ 
G) $\frac { \pi } { 8 }$ 
H) $\frac { 46 } { 27 }$

Question 5

The average value of f(x, y) over a region D in the plane with area A(D) is $\frac { 1 } { A ( D ) } \iint _ { D } f ( x , y ) d A$ . Find the average value of f(x, y) = xy over the region $D = \left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 1 , x y \geq 0 \right\}$ .

Accepted Answer

The answer of The average value of f(x, y) over...

Question 6

Calculate the double Riemann sum of f for the partition of R given by the indicated lines and the given choice of $\left( x _ { i j } ^ { * } , y _ { i j } ^ { * } \right)$ . $f ( x , y ) = x ^ { 2 } + 4 y$ , $R = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq 3 \}$ , x = 1, y = 1, y = 2; $\left( x _ { i j } ^ { * } , y _ { i j } ^ { * } \right)$ = center of Rij.

Accepted Answer

The answer of Calculate the double Riemann sum of f...

Question 7

Evaluate $\iiint _ { E } z \left( x ^ { 2 } + y ^ { 2 } \right) d V$ , where E is the solid bounded by the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ , above by z = 3 and below by z = 0.

Accepted Answer

The answer of Evaluate \(\iiint _ { E } z...

Question 8

Write $\iint _ { R } f ( x , y ) d A$ as an iterated integral in polar coordinates, where R is the region shown below.

Accepted Answer

The answer of Write \(\iint _ { R } f...

Question 9

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } d x d y$ .

Accepted Answer

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

H)1 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 1 } { 3 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  $\frac { 3 } { 4 }$ 
E)  $\frac { 2 } { 3 }$
F) $\frac { 1 } { 6 }$ 
 
G) $\frac { 5 } { 6 }$

H)1

Question 10

Evaluate the iterated integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { x ^ { 2 } } \int _ { 0 } ^ { \mathrm { In  } x } x e ^ { y } d y d z d x$ .

Accepted Answer

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

Question 11

A region W in $\mathbb { R } ^ { 3 }$ is described completely by $x \geq 0$ , $y \geq 0$ , $z \geq 0$ , and $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 4$ .(a) Describe or sketch this region.   (b) Write an integral in rectangular coordinates which gives the volume of W. Do not work out this integral.(c) Write an integral in spherical coordinates which gives the volume of W. Find the volume of W using this integral.

Accepted Answer

The answer of A region W in \(\mathbb { R...

Question 12

Compute the Riemann sum for the double integral $\iint _ { R } x + 2 y d A$ where $R = [ 0,6 ] \times [ 0,2 ]$ for the given grid and choice of sample points.

Accepted Answer

The answer of Compute the Riemann sum for the double...

Question 13

Evaluate $\int _ { 0 } ^ { 1 } \int _ { x ^ { 2 } } ^ { 1 } \int _ { 0 } ^ { 3 y } \left( y + 2 x ^ { 2 } z \right) d z d y d x$ .

Accepted Answer

The answer of Evaluate \(\int _ { 0 } ^...

Question 14

Let T be the transformation given by x = 2u + v, y = 3u.(a) A region S in the uv-plane is given below. Sketch the image R of S in the xy-plane.   (b) Find the inverse transformation $T ^ { - 1 }$ .(c) Evaluate the double integral $\iint _ { R } e ^ { y / ( 3 x - 2 y ) } d A$ .

Accepted Answer

The answer of Let T be the transformation given by...

Question 15

Find the volume under the paraboloid $z = 4 x ^ { 2 } + y ^ { 2 }$ above the triangle with vertices (0, 0, 0), (3, 0, 0), and (3, 1, 0).

Accepted Answer

A)  $\frac { 109 } { 4 }$ 
B)  $\frac { 55 } { 2 }$ 
C)  $\frac { 111 } { 4 }$ 
D) 28 
E)  $\frac { 113 } { 4 }$
F) $\frac { 57 } { 2 }$ 
 
G) $\frac { 115 } { 4 }$ 
H)29 
A)  $\frac { 109 } { 4 }$ 
B)  $\frac { 55 } { 2 }$ 
C)  $\frac { 111 } { 4 }$ 
D) 28 
E)  $\frac { 113 } { 4 }$
F) $\frac { 57 } { 2 }$ 
 
G) $\frac { 115 } { 4 }$ 
H)29

Question 16

Sketch the solid whose volume is given by the iterated integral $$\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 3 } ( 5 - x - y ) d y d x$$ .

Accepted Answer

The answer of Sketch the solid whose volume is given...

Question 17

Find the area of that part of the plane $2 x + 3 y - z + 1 = 0$ that lies above the rectangle [1, 4] $\times$ [2, 4].

Accepted Answer

The answer of Find the area of that part of...

Question 18

Find the area inside the circle $r = 2 \cos \theta$ and outside the circle r = 1.

Accepted Answer

The answer of Find the area inside the circle \(r...

Question 19

A phonograph turntable is made in the shape of a circular disk of radius 6 inches with density function $p ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ . Find the mass of the disk.

Accepted Answer

A) 60$\pi$ 
B) 96$\pi$ 
C) 108$\pi$ 
D) 120$\pi$ 
E) 144$\pi$
F)180$\pi$
G)192$\pi$
H)240$\pi$ 
A) 60$\pi$ 
B) 96$\pi$ 
C) 108$\pi$ 
D) 120$\pi$ 
E) 144$\pi$
F)180$\pi$
G)192$\pi$
H)240$\pi$

Question 20

Write $\iint _ { R } f ( x , y ) d A$ as an iterated integral in polar coordinates, where R is the region shown below.

Accepted Answer

The answer of Write \(\iint _ { R } f...

Find the Jacobian of the transformation $x = u ^ { 2 }$ , $y = v ^ { 3 }$ .

Find $\int _ { 0 } ^ { 2 } f ( x , y ) d y$ and $\int _ { 0 } ^ { 1 } f ( x , y ) d x$ for $f ( x , y ) = 2 x y - 3 x ^ { 2 }$ .

A solid is bounded above by the paraboloid $z = 1 - x ^ { 2 } - y ^ { 2 }$ and below by the xy-plane. Compute the volume of this solid using polar coordinates.

Use a double integral to find the area of the region enclosed by one loop of the curve $r = \cos 2 \theta$ .

The average value of f(x, y) over a region D in the plane with area A(D) is $\frac { 1 } { A ( D ) } \iint _ { D } f ( x , y ) d A$ . Find the average value of f(x, y) = xy over the region $D = \left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 1 , x y \geq 0 \right\}$ .

Evaluate $\iiint _ { E } z \left( x ^ { 2 } + y ^ { 2 } \right) d V$ , where E is the solid bounded by the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ , above by z = 3 and below by z = 0.

Write $\iint _ { R } f ( x , y ) d A$ as an iterated integral in polar coordinates, where R is the region shown below. $Write \iint _ { R } f ( x , y ) d A as an iterated integral in polar coordinates, where R is the region shown below.$

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } d x d y$ .

Evaluate the iterated integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { x ^ { 2 } } \int _ { 0 } ^ { \mathrm { In } x } x e ^ { y } d y d z d x$ .

Evaluate $\int _ { 0 } ^ { 1 } \int _ { x ^ { 2 } } ^ { 1 } \int _ { 0 } ^ { 3 y } \left( y + 2 x ^ { 2 } z \right) d z d y d x$ .

Find the volume under the paraboloid $z = 4 x ^ { 2 } + y ^ { 2 }$ above the triangle with vertices (0, 0, 0), (3, 0, 0), and (3, 1, 0).

Sketch the solid whose volume is given by the iterated integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 3 } ( 5 - x - y ) d y d x$ .

Find the area of that part of the plane $2 x + 3 y - z + 1 = 0$ that lies above the rectangle [1, 4] $\times$ [2, 4].

Find the area inside the circle $r = 2 \cos \theta$ and outside the circle r = 1.

A phonograph turntable is made in the shape of a circular disk of radius 6 inches with density function $p ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ . Find the mass of the disk.

Write $\iint _ { R } f ( x , y ) d A$ as an iterated integral in polar coordinates, where R is the region shown below. $Write \iint _ { R } f ( x , y ) d A as an iterated integral in polar coordinates, where R is the region shown below.$

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Exam 12: Multiple Integrals

Find the Jacobian of the transformation x=u2x = u ^ { 2 }x=u2 , y=v3y = v ^ { 3 }y=v3 .

Find ∫02f(x,y)dy\int _ { 0 } ^ { 2 } f ( x , y ) d y∫02​f(x,y)dy and ∫01f(x,y)dx\int _ { 0 } ^ { 1 } f ( x , y ) d x∫01​f(x,y)dx for f(x,y)=2xy−3x2f ( x , y ) = 2 x y - 3 x ^ { 2 }f(x,y)=2xy−3x2 .

A solid is bounded above by the paraboloid z=1−x2−y2z = 1 - x ^ { 2 } - y ^ { 2 }z=1−x2−y2 and below by the xy-plane. Compute the volume of this solid using polar coordinates.

Use a double integral to find the area of the region enclosed by one loop of the curve r=cos⁡2θr = \cos 2 \thetar=cos2θ .

Evaluate ∭Ez(x2+y2)dV\iiint _ { E } z \left( x ^ { 2 } + y ^ { 2 } \right) d V∭E​z(x2+y2)dV , where E is the solid bounded by the cylinder x2+y2=4x ^ { 2 } + y ^ { 2 } = 4x2+y2=4 , above by z = 3 and below by z = 0.

Write ∬Rf(x,y)dA\iint _ { R } f ( x , y ) d A∬R​f(x,y)dA as an iterated integral in polar coordinates, where R is the region shown below.

Evaluate the iterated integral ∫01∫0ydxdy\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } d x d y∫01​∫0y​dxdy .

Evaluate the iterated integral ∫02∫0x2∫0Inxxeydydzdx\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { x ^ { 2 } } \int _ { 0 } ^ { \mathrm { In } x } x e ^ { y } d y d z d x∫02​∫0x2​∫0Inx​xeydydzdx .

Compute the Riemann sum for the double integral ∬Rx+2ydA\iint _ { R } x + 2 y d A∬R​x+2ydA where R=[0,6]×[0,2]R = [ 0,6 ] \times [ 0,2 ]R=[0,6]×[0,2] for the given grid and choice of sample points.

Evaluate ∫01∫x21∫03y(y+2x2z)dzdydx\int _ { 0 } ^ { 1 } \int _ { x ^ { 2 } } ^ { 1 } \int _ { 0 } ^ { 3 y } \left( y + 2 x ^ { 2 } z \right) d z d y d x∫01​∫x21​∫03y​(y+2x2z)dzdydx .

Find the volume under the paraboloid z=4x2+y2z = 4 x ^ { 2 } + y ^ { 2 }z=4x2+y2 above the triangle with vertices (0, 0, 0), (3, 0, 0), and (3, 1, 0).

Sketch the solid whose volume is given by the iterated integral ∫02∫03(5−x−y)dydx\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 3 } ( 5 - x - y ) d y d x∫02​∫03​(5−x−y)dydx .

Find the area of that part of the plane 2x+3y−z+1=02 x + 3 y - z + 1 = 02x+3y−z+1=0 that lies above the rectangle [1, 4] ×\times× [2, 4].

Find the area inside the circle r=2cos⁡θr = 2 \cos \thetar=2cosθ and outside the circle r = 1.

A phonograph turntable is made in the shape of a circular disk of radius 6 inches with density function p(x,y)=x2+y2p ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }p(x,y)=x2+y2​ . Find the mass of the disk.

Write ∬Rf(x,y)dA\iint _ { R } f ( x , y ) d A∬R​f(x,y)dA as an iterated integral in polar coordinates, where R is the region shown below.

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Find the Jacobian of the transformation $x = u ^ { 2 }$ , $y = v ^ { 3 }$ .

Find $\int _ { 0 } ^ { 2 } f ( x , y ) d y$ and $\int _ { 0 } ^ { 1 } f ( x , y ) d x$ for $f ( x , y ) = 2 x y - 3 x ^ { 2 }$ .

A solid is bounded above by the paraboloid $z = 1 - x ^ { 2 } - y ^ { 2 }$ and below by the xy-plane. Compute the volume of this solid using polar coordinates.

Use a double integral to find the area of the region enclosed by one loop of the curve $r = \cos 2 \theta$ .

Evaluate $\iiint _ { E } z \left( x ^ { 2 } + y ^ { 2 } \right) d V$ , where E is the solid bounded by the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ , above by z = 3 and below by z = 0.

Write $\iint _ { R } f ( x , y ) d A$ as an iterated integral in polar coordinates, where R is the region shown below. $Write \iint _ { R } f ( x , y ) d A as an iterated integral in polar coordinates, where R is the region shown below.$

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } d x d y$ .

Evaluate the iterated integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { x ^ { 2 } } \int _ { 0 } ^ { \mathrm { In } x } x e ^ { y } d y d z d x$ .

Evaluate $\int _ { 0 } ^ { 1 } \int _ { x ^ { 2 } } ^ { 1 } \int _ { 0 } ^ { 3 y } \left( y + 2 x ^ { 2 } z \right) d z d y d x$ .

Find the volume under the paraboloid $z = 4 x ^ { 2 } + y ^ { 2 }$ above the triangle with vertices (0, 0, 0), (3, 0, 0), and (3, 1, 0).

Sketch the solid whose volume is given by the iterated integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 3 } ( 5 - x - y ) d y d x$ .

Find the area of that part of the plane $2 x + 3 y - z + 1 = 0$ that lies above the rectangle [1, 4] $\times$ [2, 4].

Find the area inside the circle $r = 2 \cos \theta$ and outside the circle r = 1.

A phonograph turntable is made in the shape of a circular disk of radius 6 inches with density function $p ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ . Find the mass of the disk.

Write $\iint _ { R } f ( x , y ) d A$ as an iterated integral in polar coordinates, where R is the region shown below. $Write \iint _ { R } f ( x , y ) d A as an iterated integral in polar coordinates, where R is the region shown below.$