Question 1

Change the order of the integration of the integral $\int _ { 0 } ^ { 1 } \int _ { e } ^ { e } f ( x , y ) d x d y$ .

Accepted Answer

The answer of Change the order of the integration of...

Question 2

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

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 3

Evaluate $\int _ { 0 } ^ { \ln 2 } \int _ { 0 } ^ { \ln 3 } e ^ { 2 x + y } d y d x$ .

Accepted Answer

A) 18 
B) 9 
C) 3 
D) ln 3 
E) 6
F)12
G)1.5
H)ln 2 
A) 18 
B) 9 
C) 3 
D) ln 3 
E) 6
F)12
G)1.5
H)ln 2

Question 4

Find the mass and center of mass of the lamina that occupies the region $D = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq 3 \}$ and has density function $p ( x , y ) = y$ .

Accepted Answer

m = 9, (x,

Question 5

Express the integral $\int _ { 0 } ^ { 2 } \int _ { x ^ { 2 } } ^ { 4 } \left( x y ^ { 2 } + x \right) d y d x$ as an equivalent integral with the order of integration reversed.

Accepted Answer

The answer of Express the integral \(\int _ { 0...

Question 6

Find the volume of the solid under the surface $z = x ^ { 2 } + y ^ { 2 }$ and lying above the region $\left\{ ( x , y ) \mid 0 \leq x \leq 1 , x ^ { 2 } \leq y \leq \sqrt { x } \right\}$ .

Accepted Answer

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

Question 7

Find the Jacobian of the transformation x = u, y = 2v, z = 3w.

Accepted Answer

A) 1 
B) 2 
C) 3 
D) 6 
E) -1
F)-2
G)-3
H)-6 
A) 1 
B) 2 
C) 3 
D) 6 
E) -1
F)-2
G)-3
H)-6

Question 8

Evaluate $\iint _ { R } 3 d A$ where $R = [ - 1,1 ] \times [ 2,3 ]$ by first identifying it as the volume of a solid.

Accepted Answer

The answer of Evaluate \(\iint _ { R } 3...

Question 9

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 2 } ( x + y ) d x d y$ .

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B) 2 
C)  $\frac { 3 } { 2 }$ 
D) 2. 
E)  $\frac { 5 } { 2 }$
F)3
G) $\frac { 7 } { 2 }$

H)4 
A)  $\frac { 1 } { 2 }$ 
B) 2 
C)  $\frac { 3 } { 2 }$ 
D) 2. 
E)  $\frac { 5 } { 2 }$
F)3
G) $\frac { 7 } { 2 }$

H)4

Question 10

Evaluate $\iint _ { R } \frac { \sin ( x + y ) } { \cos ( x - y ) } d A$ ,where R is the rectangular region bounded by the lines x + y = 0, $x + y = \frac { \pi } { 2 }$ , x - y = 0, and $x - y = \frac { \pi } { 4 }$ .

Accepted Answer

The answer of Evaluate \(\iint _ { R } \frac...

Question 11

Find a transformation x = x (u, v), y = y (u, v) maps the region in the uv-plane into the xy-plane.

Accepted Answer

The answer of Find a transformation x = x (u,...

Question 12

Give a geometric description of the solid S whose volume in cylindrical coordinates is given by $V = \int _ { 0 } ^ { 2 x } \int _ { 1 } ^ { 2 } \int _ { - \sqrt { 3 } } ^ { \sqrt { 3 } } r d z d r d \theta$ .

Accepted Answer

A cylinder of radius 2 and hei

Question 13

Find the volume of the solid obtained by intersecting the two cones $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ and $z = 1 - \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Accepted Answer

A)  $\frac { \pi } { 12 }$ 
B)  $\frac { \pi } { 6 }$ 
C)  $\frac { \pi } { 4 }$ 
D)  $\frac { \pi } { 3 }$ 
E)  $\frac { \pi } { 2 }$

F)$\frac { 3 \pi } { 4 }$

G) $\frac { 5 \pi } { 6 }$

H)$\pi$ 
A)  $\frac { \pi } { 12 }$ 
B)  $\frac { \pi } { 6 }$ 
C)  $\frac { \pi } { 4 }$ 
D)  $\frac { \pi } { 3 }$ 
E)  $\frac { \pi } { 2 }$

F)$\frac { 3 \pi } { 4 }$

G) $\frac { 5 \pi } { 6 }$

H)$\pi$

Question 14

Find the moment of inertia $I _ { y }$ of the lamina that occupies the region $D = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ and whose density function at any point is the distance from that point to the x-axis.

Accepted Answer

A)  $\frac { 1 } { 24 }$ 
B)  $\frac { 1 } { 18 }$ 
C)  $\frac { 1 } { 12 }$ 
D)  $\frac { 1 } { 9 }$ 
E)  $\frac { 1 } { 6 }$
F) $\frac { 1 } { 4 }$

G) $\frac { 1 } { 3 }$

H) $\frac { 1 } { 2 }$ 
A)  $\frac { 1 } { 24 }$ 
B)  $\frac { 1 } { 18 }$ 
C)  $\frac { 1 } { 12 }$ 
D)  $\frac { 1 } { 9 }$ 
E)  $\frac { 1 } { 6 }$
F) $\frac { 1 } { 4 }$

G) $\frac { 1 } { 3 }$

H) $\frac { 1 } { 2 }$

Question 15

Evaluate $\iiint _ { z } e ^ { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } } d V$ , where E is the solid bounded by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ .

Accepted Answer

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

Question 16

Evaluate $\int _ { 0 } ^ { 1 } \int _ { x + 1 } ^ { 2 } e ^ { x / ( y - 1 ) } d y d x$ .

Accepted Answer

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

Question 17

Let $f ( x , y ) = x$ , and let $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ . Let R be partitioned into two subrectangles by the line $x = \frac { 1 } { 4 }$ , and let $\left( x _ { i } ^ { * } , y _ { j } ^ { * } \right)$ be the center of Rij. Calculate the double Riemann sum of f.

Accepted Answer

A)  $\frac { 3 } { 16 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 5 } { 16 }$ 
D)  $\frac { 3 } { 8 }$ 
E)  $\frac { 7 } { 16 }$  
F) $\frac { 1 } { 2 }$ 
 
G) $\frac { 9 } { 16 }$

H) $\frac { 3 } { 4 }$ 
A)  $\frac { 3 } { 16 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 5 } { 16 }$ 
D)  $\frac { 3 } { 8 }$ 
E)  $\frac { 7 } { 16 }$  
F) $\frac { 1 } { 2 }$ 
 
G) $\frac { 9 } { 16 }$

H) $\frac { 3 } { 4 }$

Question 18

Evaluate $\int _ { 0 } ^ { 1 } \int _ { 2 x } ^ { 2 } \frac { 1 } { \sqrt { 1 + ( 2 x / y ) } } d y d x$ .

Accepted Answer

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

Question 19

Find the volume of the solid obtained by intersecting the two paraboloids $z = x ^ { 2 } + y ^ { 2 }$ and $z = 1 - x ^ { 2 } - y ^ { 2 }$ .

Accepted Answer

A)  $\frac { \pi } { 6 }$ 
B)  $\frac { \pi } { 4 }$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { \pi } { 2 }$ 
E)  $\frac { 2 \pi } { 3 }$

F) $\frac { 3 \pi } { 4 }$

G) $\frac { 5 \pi } { 6 }$

H)$\pi$ 
A)  $\frac { \pi } { 6 }$ 
B)  $\frac { \pi } { 4 }$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { \pi } { 2 }$ 
E)  $\frac { 2 \pi } { 3 }$

F) $\frac { 3 \pi } { 4 }$

G) $\frac { 5 \pi } { 6 }$

H)$\pi$

Question 20

Evaluate the following double integral: $\int _ { 0 } ^ { a } \int _ { 0 } ^ { \sqrt { e ^ { 2 } - x ^ { 2 } } } ( x + y ) d y d x$ , where a > 0.

Accepted Answer

The answer of Evaluate the following double integral: \(\int _...

Change the order of the integration of the integral $\int _ { 0 } ^ { 1 } \int _ { e } ^ { e } f ( x , y ) d x d y$ .

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

Evaluate $\int _ { 0 } ^ { \ln 2 } \int _ { 0 } ^ { \ln 3 } e ^ { 2 x + y } d y d x$ .

Find the mass and center of mass of the lamina that occupies the region $D = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq 3 \}$ and has density function $p ( x , y ) = y$ .

Express the integral $\int _ { 0 } ^ { 2 } \int _ { x ^ { 2 } } ^ { 4 } \left( x y ^ { 2 } + x \right) d y d x$ as an equivalent integral with the order of integration reversed.

Find the volume of the solid under the surface $z = x ^ { 2 } + y ^ { 2 }$ and lying above the region $\left\{ ( x , y ) \mid 0 \leq x \leq 1 , x ^ { 2 } \leq y \leq \sqrt { x } \right\}$ .

Find the Jacobian of the transformation x = u, y = 2v, z = 3w.

Evaluate $\iint _ { R } 3 d A$ where $R = [ - 1,1 ] \times [ 2,3 ]$ by first identifying it as the volume of a solid.

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 2 } ( x + y ) d x d y$ .

Evaluate $\iint _ { R } \frac { \sin ( x + y ) } { \cos ( x - y ) } d A$ ,where R is the rectangular region bounded by the lines x + y = 0, $x + y = \frac { \pi } { 2 }$ , x - y = 0, and $x - y = \frac { \pi } { 4 }$ .

Find a transformation x = x (u, v), y = y (u, v) maps the region in the uv-plane into the xy-plane.

Give a geometric description of the solid S whose volume in cylindrical coordinates is given by $V = \int _ { 0 } ^ { 2 x } \int _ { 1 } ^ { 2 } \int _ { - \sqrt { 3 } } ^ { \sqrt { 3 } } r d z d r d \theta$ .

Find the volume of the solid obtained by intersecting the two cones $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ and $z = 1 - \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Find the moment of inertia $I _ { y }$ of the lamina that occupies the region $D = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ and whose density function at any point is the distance from that point to the x-axis.

Evaluate $\iiint _ { z } e ^ { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } } d V$ , where E is the solid bounded by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ .

Evaluate $\int _ { 0 } ^ { 1 } \int _ { x + 1 } ^ { 2 } e ^ { x / ( y - 1 ) } d y d x$ .

Let $f ( x , y ) = x$ , and let $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ . Let R be partitioned into two subrectangles by the line $x = \frac { 1 } { 4 }$ , and let $\left( x _ { i } ^ { * } , y _ { j } ^ { * } \right)$ be the center of Rij. Calculate the double Riemann sum of f.

Evaluate $\int _ { 0 } ^ { 1 } \int _ { 2 x } ^ { 2 } \frac { 1 } { \sqrt { 1 + ( 2 x / y ) } } d y d x$ .

Find the volume of the solid obtained by intersecting the two paraboloids $z = x ^ { 2 } + y ^ { 2 }$ and $z = 1 - x ^ { 2 } - y ^ { 2 }$ .

Evaluate the following double integral: $\int _ { 0 } ^ { a } \int _ { 0 } ^ { \sqrt { e ^ { 2 } - x ^ { 2 } } } ( x + y ) d y d x$ , where a > 0.

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Exam 12: Multiple Integrals

Change the order of the integration of the integral ∫01∫eef(x,y)dxdy\int _ { 0 } ^ { 1 } \int _ { e } ^ { e } f ( x , y ) d x d y∫01​∫ee​f(x,y)dxdy .

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

Evaluate ∫0ln⁡2∫0ln⁡3e2x+ydydx\int _ { 0 } ^ { \ln 2 } \int _ { 0 } ^ { \ln 3 } e ^ { 2 x + y } d y d x∫0ln2​∫0ln3​e2x+ydydx .

Find the mass and center of mass of the lamina that occupies the region D={(x,y)∣0≤x≤2,0≤y≤3}D = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq 3 \}D={(x,y)∣0≤x≤2,0≤y≤3} and has density function p(x,y)=yp ( x , y ) = yp(x,y)=y .

Express the integral ∫02∫x24(xy2+x)dydx\int _ { 0 } ^ { 2 } \int _ { x ^ { 2 } } ^ { 4 } \left( x y ^ { 2 } + x \right) d y d x∫02​∫x24​(xy2+x)dydx as an equivalent integral with the order of integration reversed.

Find the volume of the solid under the surface z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 and lying above the region {(x,y)∣0≤x≤1,x2≤y≤x}\left\{ ( x , y ) \mid 0 \leq x \leq 1 , x ^ { 2 } \leq y \leq \sqrt { x } \right\}{(x,y)∣0≤x≤1,x2≤y≤x​} .

Find the Jacobian of the transformation x = u, y = 2v, z = 3w.

Evaluate ∬R3dA\iint _ { R } 3 d A∬R​3dA where R=[−1,1]×[2,3]R = [ - 1,1 ] \times [ 2,3 ]R=[−1,1]×[2,3] by first identifying it as the volume of a solid.

Evaluate the iterated integral ∫01∫02(x+y)dxdy\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 2 } ( x + y ) d x d y∫01​∫02​(x+y)dxdy .

Find a transformation x = x (u, v), y = y (u, v) maps the region in the uv-plane into the xy-plane.

Give a geometric description of the solid S whose volume in cylindrical coordinates is given by V=∫02x∫12∫−33rdzdrdθV = \int _ { 0 } ^ { 2 x } \int _ { 1 } ^ { 2 } \int _ { - \sqrt { 3 } } ^ { \sqrt { 3 } } r d z d r d \thetaV=∫02x​∫12​∫−3​3​​rdzdrdθ .

Find the volume of the solid obtained by intersecting the two cones z=x2+y2z = \sqrt { x ^ { 2 } + y ^ { 2 } }z=x2+y2​ and z=1−x2+y2z = 1 - \sqrt { x ^ { 2 } + y ^ { 2 } }z=1−x2+y2​ .

Find the moment of inertia IyI _ { y }Iy​ of the lamina that occupies the region D={(x,y)∣0≤x≤1,0≤y≤1}D = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}D={(x,y)∣0≤x≤1,0≤y≤1} and whose density function at any point is the distance from that point to the x-axis.

Evaluate ∭ze(x2+y2+z2)3/2dV\iiint _ { z } e ^ { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } } d V∭z​e(x2+y2+z2)3/2dV , where E is the solid bounded by the sphere x2+y2+z2=1x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1x2+y2+z2=1 .

Evaluate ∫01∫x+12ex/(y−1)dydx\int _ { 0 } ^ { 1 } \int _ { x + 1 } ^ { 2 } e ^ { x / ( y - 1 ) } d y d x∫01​∫x+12​ex/(y−1)dydx .

Evaluate ∫01∫2x211+(2x/y)dydx\int _ { 0 } ^ { 1 } \int _ { 2 x } ^ { 2 } \frac { 1 } { \sqrt { 1 + ( 2 x / y ) } } d y d x∫01​∫2x2​1+(2x/y)​1​dydx .

Find the volume of the solid obtained by intersecting the two paraboloids z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2 and z=1−x2−y2z = 1 - x ^ { 2 } - y ^ { 2 }z=1−x2−y2 .

Evaluate the following double integral: ∫0a∫0e2−x2(x+y)dydx\int _ { 0 } ^ { a } \int _ { 0 } ^ { \sqrt { e ^ { 2 } - x ^ { 2 } } } ( x + y ) d y d x∫0a​∫0e2−x2​​(x+y)dydx , where a > 0.

Functions and Models

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Vectors and the Geometry of Space

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Change the order of the integration of the integral $\int _ { 0 } ^ { 1 } \int _ { e } ^ { e } f ( x , y ) d x d y$ .

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

Evaluate $\int _ { 0 } ^ { \ln 2 } \int _ { 0 } ^ { \ln 3 } e ^ { 2 x + y } d y d x$ .

Find the mass and center of mass of the lamina that occupies the region $D = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq 3 \}$ and has density function $p ( x , y ) = y$ .

Express the integral $\int _ { 0 } ^ { 2 } \int _ { x ^ { 2 } } ^ { 4 } \left( x y ^ { 2 } + x \right) d y d x$ as an equivalent integral with the order of integration reversed.

Find the volume of the solid under the surface $z = x ^ { 2 } + y ^ { 2 }$ and lying above the region $\left\{ ( x , y ) \mid 0 \leq x \leq 1 , x ^ { 2 } \leq y \leq \sqrt { x } \right\}$ .

Evaluate $\iint _ { R } 3 d A$ where $R = [ - 1,1 ] \times [ 2,3 ]$ by first identifying it as the volume of a solid.

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 2 } ( x + y ) d x d y$ .

Give a geometric description of the solid S whose volume in cylindrical coordinates is given by $V = \int _ { 0 } ^ { 2 x } \int _ { 1 } ^ { 2 } \int _ { - \sqrt { 3 } } ^ { \sqrt { 3 } } r d z d r d \theta$ .

Find the volume of the solid obtained by intersecting the two cones $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ and $z = 1 - \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Find the moment of inertia $I _ { y }$ of the lamina that occupies the region $D = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ and whose density function at any point is the distance from that point to the x-axis.

Evaluate $\iiint _ { z } e ^ { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } } d V$ , where E is the solid bounded by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ .

Evaluate $\int _ { 0 } ^ { 1 } \int _ { x + 1 } ^ { 2 } e ^ { x / ( y - 1 ) } d y d x$ .

Evaluate $\int _ { 0 } ^ { 1 } \int _ { 2 x } ^ { 2 } \frac { 1 } { \sqrt { 1 + ( 2 x / y ) } } d y d x$ .

Find the volume of the solid obtained by intersecting the two paraboloids $z = x ^ { 2 } + y ^ { 2 }$ and $z = 1 - x ^ { 2 } - y ^ { 2 }$ .

Evaluate the following double integral: $\int _ { 0 } ^ { a } \int _ { 0 } ^ { \sqrt { e ^ { 2 } - x ^ { 2 } } } ( x + y ) d y d x$ , where a > 0.