Question 1

Let $\bar { y }$ denote the y-coordinate of the centroid of the region enclosed by $y = x ^ { 2 }$ and $y = 1$ Then $\bar { y }$ =

Accepted Answer

A)  $\frac { 1,125 } { 8 }$ 
B)  $\frac { 3 } { 5 }$ 
C) 1 
D)  $\frac { 3 } { 4 }$ 
E)  $\frac { 45 } { 28 }$ 
A)  $\frac { 1,125 } { 8 }$ 
B)  $\frac { 3 } { 5 }$ 
C) 1 
D)  $\frac { 3 } { 4 }$ 
E)  $\frac { 45 } { 28 }$

Question 2

The arc length of $y = ( x - 3 ) ^ { \frac { 3 } { 2 } } - 1 \text { for } x \in [ 3,6 ]$ is

Accepted Answer

A)  $\frac { 31 \sqrt { 31 } - 8 } { 27 }$ 
B)  $\frac { 31 \sqrt { 31 } - 13 \sqrt { 13 } } { 27 }$ 
C)  $\frac { 31 \sqrt { 31 } - 12 \sqrt { 13 } } { 27 }$ 
D)  $\frac { 31 \sqrt { 31 } + 13 \sqrt { 13 } } { 27 }$ 
E)  $\frac { 31 \sqrt { 31 } - 17 \sqrt { 17 } } { 27 }$ 
A)  $\frac { 31 \sqrt { 31 } - 8 } { 27 }$ 
B)  $\frac { 31 \sqrt { 31 } - 13 \sqrt { 13 } } { 27 }$ 
C)  $\frac { 31 \sqrt { 31 } - 12 \sqrt { 13 } } { 27 }$ 
D)  $\frac { 31 \sqrt { 31 } + 13 \sqrt { 13 } } { 27 }$ 
E)  $\frac { 31 \sqrt { 31 } - 17 \sqrt { 17 } } { 27 }$

Question 3

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = -3 to x = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = - \sqrt { 9 - x ^ { 2 } }$ to $y = \sqrt { 9 - x ^ { 2 } }$ and they are equilateral triangles with bases in the xy-plane. Then V is

Accepted Answer

A)  $18 \sqrt { 2 }$ 
B)  $36 \sqrt { 3 }$ 
C) 36 
D)  $18 \sqrt { 5 }$ 
E) 9 
A)  $18 \sqrt { 2 }$ 
B)  $36 \sqrt { 3 }$ 
C) 36 
D)  $18 \sqrt { 5 }$ 
E) 9

Question 4

Let $\bar { y }$ denote the y-coordinate of the centroid of the region enclosed by $y = x ^ { 2 }$ and $y = 2 x + 3$ Then $\bar { y }$ =

Accepted Answer

A)  $\frac { 1,125 } { 8 }$ 
B)  $\frac { 17 } { 5 }$ 
C)  $\frac { 93 } { 35 }$ 
D)  $\frac { 3 } { 4 }$ 
E)  $\frac { 45 } { 28 }$ 
A)  $\frac { 1,125 } { 8 }$ 
B)  $\frac { 17 } { 5 }$ 
C)  $\frac { 93 } { 35 }$ 
D)  $\frac { 3 } { 4 }$ 
E)  $\frac { 45 } { 28 }$

Question 5

Let V be the volume of the solid generated by revolving the region enclosed by $y = \sqrt { x },$ the x-axis, $x = 1;$ about the y-axis. Then V is

Accepted Answer

A)  $\frac { 4 \pi } { 5 }$ 
B)  $\frac { 8 \pi } { 5 }$ 
C)  $\frac { 16 \pi } { 5 }$ 
D)  $\frac { 32 \pi } { 5 }$ 
E)  $\frac { 64 \pi } { 5 }$ 
A)  $\frac { 4 \pi } { 5 }$ 
B)  $\frac { 8 \pi } { 5 }$ 
C)  $\frac { 16 \pi } { 5 }$ 
D)  $\frac { 32 \pi } { 5 }$ 
E)  $\frac { 64 \pi } { 5 }$

Question 6

Let V be the volume of the solid generated by revolving the region enclosed by $y = 2 | x | \text {, }$ the x-axis, $x = - 1 ;$ about the x-axis. Then V is

Accepted Answer

A)  $\frac { 13 \pi } { 3 }$ 
B)  $\frac { 11 \pi } { 3 }$ 
C)  $\frac { 10 \pi } { 3 }$ 
D)  $\frac { 4 \pi } { 3 }$ 
E)  $\frac { 7 \pi } { 3 }$ 
A)  $\frac { 13 \pi } { 3 }$ 
B)  $\frac { 11 \pi } { 3 }$ 
C)  $\frac { 10 \pi } { 3 }$ 
D)  $\frac { 4 \pi } { 3 }$ 
E)  $\frac { 7 \pi } { 3 }$

Question 7

A hemispherical water tank of radius 6 meters is filled with water to a depth of 4 meters. Assume the density of water is 1000 kg/cu m. The work required to pump all the water over the top of the tank, in kg-meters, is

Accepted Answer

A)  $2,805,800 \pi$ 
B)  $2,708,800 \pi$ 
C)  $2,635,800 \pi$ 
D)  $2,619,400 \pi$ 
E)  $2,508,800 \pi$ 
A)  $2,805,800 \pi$ 
B)  $2,708,800 \pi$ 
C)  $2,635,800 \pi$ 
D)  $2,619,400 \pi$ 
E)  $2,508,800 \pi$

Question 8

The arc length of $y = \frac { x ^ { 3 } } { 6 } + \frac { 1 } { 2 x } \text { for } x \in [ 2,5 ]$ is

Accepted Answer

A)  $\frac { 393 } { 20 }$ 
B) 45 
C)  $\frac { 393 } { 16 }$ 
D)  $\frac { 263 } { 16 }$ 
E)  $\frac { 225 } { 16 }$ 
A)  $\frac { 393 } { 20 }$ 
B) 45 
C)  $\frac { 393 } { 16 }$ 
D)  $\frac { 263 } { 16 }$ 
E)  $\frac { 225 } { 16 }$

Question 9

A right circular cylindrical storage tank with height 10 meters and radius 8 meters is full of gasoline. Assume the density of gasoline is 720 kg/cu m. The work required to pump all the gasoline over the top of the tank, in kg-meters, is

Accepted Answer

A)  $22,897,200 \pi$ 
B)  $22,796,200 \pi$ 
C)  $22,579,200 \pi$ 
D)  $22,486,200 \pi$ 
E)  $22,388,200 \pi$ 
A)  $22,897,200 \pi$ 
B)  $22,796,200 \pi$ 
C)  $22,579,200 \pi$ 
D)  $22,486,200 \pi$ 
E)  $22,388,200 \pi$

Question 10

Let V be the volume of the solid that lies between planes perpendicular to the x-axis from x = 0 to x = 3. The cross-sections of this solid perpendicular to the x-axis run from $y = \frac { x ^ { 2 } } { 9 }$ to $y = \sqrt { \frac { x } { 3 } }$ and they are squares with bases in the xy-plane. Then V is

Accepted Answer

A)  $\frac { 27 } { 70 }$ 
B) 144 
C)  $\frac { 256 } { 3 }$ 
D)  $\frac { 72 } { 35 }$ 
E)  $\frac { 25 } { 3 }$ 
A)  $\frac { 27 } { 70 }$ 
B) 144 
C)  $\frac { 256 } { 3 }$ 
D)  $\frac { 72 } { 35 }$ 
E)  $\frac { 25 } { 3 }$

Question 11

The work done by a variable force $F ( x ) = ( 4 x - 1 ) ^ { 2 }$ Newtons that moves an object along a straight line in the direction of F from x = 2 meters to x = 4 meters, in Newton-meters, is

Accepted Answer

A)  $\frac { 50 } { 3 }$ 
B)  $\frac { 758 } { 3 }$ 
C)  $\frac { 825 } { 2 }$ 
D)  $\frac { 782 } { 27 }$ 
E)  $\frac { 302 } { 3 }$ 
A)  $\frac { 50 } { 3 }$ 
B)  $\frac { 758 } { 3 }$ 
C)  $\frac { 825 } { 2 }$ 
D)  $\frac { 782 } { 27 }$ 
E)  $\frac { 302 } { 3 }$

Question 12

The arc length of $y = x ^ { \frac { 3 } { 2 } } + 9 \text { for } x \in [ 1,4 ]$ is

Accepted Answer

A)  $\frac { 80 \sqrt { 10 } + 13 \sqrt { 13 } } { 27 }$ 
B)  $\frac { 80 \sqrt { 10 } - 12 \sqrt { 13 } } { 27 }$ 
C)  $\frac { 40 \sqrt { 10 } - 13 \sqrt { 13 } } { 27 }$ 
D)  $\frac { 80 \sqrt { 10 } - 13 \sqrt { 13 } } { 27 }$ 
E)  $\frac { 80 \sqrt { 10 } + 12 \sqrt { 13 } } { 27 }$ 
A)  $\frac { 80 \sqrt { 10 } + 13 \sqrt { 13 } } { 27 }$ 
B)  $\frac { 80 \sqrt { 10 } - 12 \sqrt { 13 } } { 27 }$ 
C)  $\frac { 40 \sqrt { 10 } - 13 \sqrt { 13 } } { 27 }$ 
D)  $\frac { 80 \sqrt { 10 } - 13 \sqrt { 13 } } { 27 }$ 
E)  $\frac { 80 \sqrt { 10 } + 12 \sqrt { 13 } } { 27 }$

Question 13

The arc length of $y = \frac { 4 } { 3 } ( x - 1 ) ^ { \frac { 3 } { 2 } } \text { for } x \in [ 2,4 ]$ is

Accepted Answer

A)  $\frac { 17 \sqrt { 17 } + 13 \sqrt { 13 } } { 6 }$ 
B)  $\frac { 17 \sqrt { 17 } - 13 \sqrt { 13 } } { 6 }$ 
C)  $\frac { 13 \sqrt { 13 } - 5 \sqrt { 5 } } { 6 }$ 
D)  $\frac { 13 \sqrt { 13 } + 5 \sqrt { 5 } } { 6 }$ 
E)  $\frac { 17 \sqrt { 17 } - 13 \sqrt { 13 } } { 7 }$ 
A)  $\frac { 17 \sqrt { 17 } + 13 \sqrt { 13 } } { 6 }$ 
B)  $\frac { 17 \sqrt { 17 } - 13 \sqrt { 13 } } { 6 }$ 
C)  $\frac { 13 \sqrt { 13 } - 5 \sqrt { 5 } } { 6 }$ 
D)  $\frac { 13 \sqrt { 13 } + 5 \sqrt { 5 } } { 6 }$ 
E)  $\frac { 17 \sqrt { 17 } - 13 \sqrt { 13 } } { 7 }$

Question 14

Four particles having masses 2, 3, 3, and 4 kg are located on the xy-plane at $( - 1 , - 2 ) , ( 1,3 ) , ( 0,5 ) , ( 2,1 ),$ respectively. The x-coordinate of the center of mass of this system is located at $\bar { x } =$ ?

Accepted Answer

A)  $\frac { 1,125 } { 8 }$ 
B)  $\frac { 1 } { 3 }$ 
C) 2 
D)  $\frac { 3 } { 4 }$ 
E)  $\frac { 125 } { 24 }$ 
A)  $\frac { 1,125 } { 8 }$ 
B)  $\frac { 1 } { 3 }$ 
C) 2 
D)  $\frac { 3 } { 4 }$ 
E)  $\frac { 125 } { 24 }$

Question 15

Let A denote the area enclosed by the equations $y = 3 - 2 | x |$ and $y = | x | \text {. }$ Then A is

Accepted Answer

A) 2 
B)  $\frac { 9 } { 4 }$ 
C)  $\frac { 11 } { 4 }$ 
D) 3 
E)  $\frac { 13 } { 4 }$ 
A) 2 
B)  $\frac { 9 } { 4 }$ 
C)  $\frac { 11 } { 4 }$ 
D) 3 
E)  $\frac { 13 } { 4 }$

Question 16

A spring in equilibrium is 0.2 meters long and an external force of 0.5 Newtons stretches the spring to a length of 0.24 meters. The work done by the spring in stretching it from equilibrium to 0.28 meters, in Newton-meters, is
​

Accepted Answer

A) -0.8 
B) -0.5 
C) -0.12 
D) -0.04 
E) -0.09 
A) -0.8 
B) -0.5 
C) -0.12 
D) -0.04 
E) -0.09

Question 17

Which of these integrals can be used to calculate the surface area of the solid formed by revolving the region bounded by $f ( x ) = \left( 1 - x ^ { 2 } \right) ^ { 3 },$ the x-axis, x = -2, and x = 0 about the x-axis?

Accepted Answer

A)  $\int _ { - 1 } ^ { 4 } \left( 1 - x ^ { 2 } \right) ^ { 3 } \sqrt { 1 + 36 x ^ { 2 } \left( 1 - x ^ { 2 } \right) ^ { 4 } } d x$ 
B)  $2 \pi \int _ { - 1 } ^ { 4 } 6 x \left( 1 - x ^ { 2 } \right) ^ { 2 } \sqrt { 1 + \left( 1 - x ^ { 2 } \right) ^ { 6 } } d x$ 
C)  $2 \pi \int _ { - 1 } ^ { 4 } \left( 1 - x ^ { 2 } \right) ^ { 3 } \sqrt { 1 + 6 x \left( 1 - x ^ { 2 } \right) ^ { 2 } } d x$ 
D)  $\int _ { - 1 } ^ { 4 } \left( 1 - x ^ { 2 } \right) ^ { 6 } \sqrt { 1 + 6 x \left( 1 - x ^ { 2 } \right) ^ { 2 } } d x$ 
E)  $2 \pi \int _ { - 1 } ^ { 4 } \left( 1 - x ^ { 2 } \right) ^ { 3 } \sqrt { 1 + 36 x ^ { 2 } \left( 1 - x ^ { 2 } \right) ^ { 4 } } d x$ 
A)  $\int _ { - 1 } ^ { 4 } \left( 1 - x ^ { 2 } \right) ^ { 3 } \sqrt { 1 + 36 x ^ { 2 } \left( 1 - x ^ { 2 } \right) ^ { 4 } } d x$ 
B)  $2 \pi \int _ { - 1 } ^ { 4 } 6 x \left( 1 - x ^ { 2 } \right) ^ { 2 } \sqrt { 1 + \left( 1 - x ^ { 2 } \right) ^ { 6 } } d x$ 
C)  $2 \pi \int _ { - 1 } ^ { 4 } \left( 1 - x ^ { 2 } \right) ^ { 3 } \sqrt { 1 + 6 x \left( 1 - x ^ { 2 } \right) ^ { 2 } } d x$ 
D)  $\int _ { - 1 } ^ { 4 } \left( 1 - x ^ { 2 } \right) ^ { 6 } \sqrt { 1 + 6 x \left( 1 - x ^ { 2 } \right) ^ { 2 } } d x$ 
E)  $2 \pi \int _ { - 1 } ^ { 4 } \left( 1 - x ^ { 2 } \right) ^ { 3 } \sqrt { 1 + 36 x ^ { 2 } \left( 1 - x ^ { 2 } \right) ^ { 4 } } d x$

Question 18

Let A denote the area enclosed by the equations $y = x ^ { 2 } + 1$ and $y = - 4 x - 2$ Then A is

Accepted Answer

A)  $\frac { 4 } { 9 }$ 
B)  $\frac { 4 } { 5 }$ 
C)  $\frac { 4 } { 3 }$ 
D)  $\frac { 8 } { 3 }$ 
E)  $\frac { 2 } { 3 }$ 
A)  $\frac { 4 } { 9 }$ 
B)  $\frac { 4 } { 5 }$ 
C)  $\frac { 4 } { 3 }$ 
D)  $\frac { 8 } { 3 }$ 
E)  $\frac { 2 } { 3 }$

Question 19

Let $\bar { y }$ denote the y-coordinate of the centroid of the region enclosed by $y ^ { 2 } = 4 x , y = 0,$ $x = 1,$ and $x = 1,$ Then $\bar { y }$ =

Accepted Answer

A)  $\frac { 1,125 } { 8 }$ 
B)  $\frac { 1 } { 3 }$ 
C)  $\frac { 93 } { 35 }$ 
D)  $\frac { 93 } { 35 }$. 
E)  $\frac { 45 } { 28 }$ 
A)  $\frac { 1,125 } { 8 }$ 
B)  $\frac { 1 } { 3 }$ 
C)  $\frac { 93 } { 35 }$ 
D)  $\frac { 93 } { 35 }$. 
E)  $\frac { 45 } { 28 }$

Question 20

Let A denote the area enclosed by the equations $y = \frac { 3 } { x }$ and $y = 4 - x$ Then A is

Accepted Answer

A) 2 
B)  $2 + \ln 3$ 
C) 3 
D)  $4 - 3 \ln 3$ 
E)  $3 + \ln 3$ 
A) 2 
B)  $2 + \ln 3$ 
C) 3 
D)  $4 - 3 \ln 3$ 
E)  $3 + \ln 3$

Let $\bar { y }$ denote the y-coordinate of the centroid of the region enclosed by $y = x ^ { 2 }$ and $y = 1$ Then $\bar { y }$ =

The arc length of $y = ( x - 3 ) ^ { \frac { 3 } { 2 } } - 1 \text { for } x \in [ 3,6 ]$ is

Let $\bar { y }$ denote the y-coordinate of the centroid of the region enclosed by $y = x ^ { 2 }$ and $y = 2 x + 3$ Then $\bar { y }$ =

Let V be the volume of the solid generated by revolving the region enclosed by $y = \sqrt { x },$ the x-axis, $x = 1;$ about the y-axis. Then V is

Let V be the volume of the solid generated by revolving the region enclosed by $y = 2 | x | \text {, }$ the x-axis, $x = - 1 ;$ about the x-axis. Then V is

A hemispherical water tank of radius 6 meters is filled with water to a depth of 4 meters. Assume the density of water is 1000 kg/cu m. The work required to pump all the water over the top of the tank, in kg-meters, is

The arc length of $y = \frac { x ^ { 3 } } { 6 } + \frac { 1 } { 2 x } \text { for } x \in [ 2,5 ]$ is

A right circular cylindrical storage tank with height 10 meters and radius 8 meters is full of gasoline. Assume the density of gasoline is 720 kg/cu m. The work required to pump all the gasoline over the top of the tank, in kg-meters, is

The work done by a variable force $F ( x ) = ( 4 x - 1 ) ^ { 2 }$ Newtons that moves an object along a straight line in the direction of F from x = 2 meters to x = 4 meters, in Newton-meters, is

The arc length of $y = x ^ { \frac { 3 } { 2 } } + 9 \text { for } x \in [ 1,4 ]$ is

The arc length of $y = \frac { 4 } { 3 } ( x - 1 ) ^ { \frac { 3 } { 2 } } \text { for } x \in [ 2,4 ]$ is

Four particles having masses 2, 3, 3, and 4 kg are located on the xy-plane at $( - 1 , - 2 ) , ( 1,3 ) , ( 0,5 ) , ( 2,1 ),$ respectively. The x-coordinate of the center of mass of this system is located at $\bar { x } =$ ?

Let A denote the area enclosed by the equations $y = 3 - 2 | x |$ and $y = | x | \text {. }$ Then A is

A spring in equilibrium is 0.2 meters long and an external force of 0.5 Newtons stretches the spring to a length of 0.24 meters. The work done by the spring in stretching it from equilibrium to 0.28 meters, in Newton-meters, is

Which of these integrals can be used to calculate the surface area of the solid formed by revolving the region bounded by $f ( x ) = \left( 1 - x ^ { 2 } \right) ^ { 3 },$ the x-axis, x = -2, and x = 0 about the x-axis?

Let A denote the area enclosed by the equations $y = x ^ { 2 } + 1$ and $y = - 4 x - 2$ Then A is

Let $\bar { y }$ denote the y-coordinate of the centroid of the region enclosed by $y ^ { 2 } = 4 x , y = 0,$ $x = 1,$ and $x = 1,$ Then $\bar { y }$ =

Let A denote the area enclosed by the equations $y = \frac { 3 } { x }$ and $y = 4 - x$ Then A is

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Exam 7: Applications of the Integral

Let yˉ\bar { y }yˉ​ denote the y-coordinate of the centroid of the region enclosed by y=x2y = x ^ { 2 }y=x2 and y=1y = 1y=1 Then yˉ\bar { y }yˉ​ =

The arc length of y=(x−3)32−1 for x∈[3,6]y = ( x - 3 ) ^ { \frac { 3 } { 2 } } - 1 \text { for } x \in [ 3,6 ]y=(x−3)23​−1 for x∈[3,6] is

Let yˉ\bar { y }yˉ​ denote the y-coordinate of the centroid of the region enclosed by y=x2y = x ^ { 2 }y=x2 and y=2x+3y = 2 x + 3y=2x+3 Then yˉ\bar { y }yˉ​ =

Let V be the volume of the solid generated by revolving the region enclosed by y=x,y = \sqrt { x },y=x​, the x-axis, x=1;x = 1;x=1; about the y-axis. Then V is

Let V be the volume of the solid generated by revolving the region enclosed by y=2∣x∣, y = 2 | x | \text {, }y=2∣x∣, the x-axis, x=−1;x = - 1 ;x=−1; about the x-axis. Then V is

A hemispherical water tank of radius 6 meters is filled with water to a depth of 4 meters. Assume the density of water is 1000 kg/cu m. The work required to pump all the water over the top of the tank, in kg-meters, is

The arc length of y=x36+12x for x∈[2,5]y = \frac { x ^ { 3 } } { 6 } + \frac { 1 } { 2 x } \text { for } x \in [ 2,5 ]y=6x3​+2x1​ for x∈[2,5] is

A right circular cylindrical storage tank with height 10 meters and radius 8 meters is full of gasoline. Assume the density of gasoline is 720 kg/cu m. The work required to pump all the gasoline over the top of the tank, in kg-meters, is

The work done by a variable force F(x)=(4x−1)2F ( x ) = ( 4 x - 1 ) ^ { 2 }F(x)=(4x−1)2 Newtons that moves an object along a straight line in the direction of F from x = 2 meters to x = 4 meters, in Newton-meters, is

The arc length of y=x32+9 for x∈[1,4]y = x ^ { \frac { 3 } { 2 } } + 9 \text { for } x \in [ 1,4 ]y=x23​+9 for x∈[1,4] is

The arc length of y=43(x−1)32 for x∈[2,4]y = \frac { 4 } { 3 } ( x - 1 ) ^ { \frac { 3 } { 2 } } \text { for } x \in [ 2,4 ]y=34​(x−1)23​ for x∈[2,4] is

Four particles having masses 2, 3, 3, and 4 kg are located on the xy-plane at (−1,−2),(1,3),(0,5),(2,1),( - 1 , - 2 ) , ( 1,3 ) , ( 0,5 ) , ( 2,1 ),(−1,−2),(1,3),(0,5),(2,1), respectively. The x-coordinate of the center of mass of this system is located at xˉ=\bar { x } =xˉ= ?

Let A denote the area enclosed by the equations y=3−2∣x∣y = 3 - 2 | x |y=3−2∣x∣ and y=∣x∣. y = | x | \text {. }y=∣x∣. Then A is

A spring in equilibrium is 0.2 meters long and an external force of 0.5 Newtons stretches the spring to a length of 0.24 meters. The work done by the spring in stretching it from equilibrium to 0.28 meters, in Newton-meters, is ​

Which of these integrals can be used to calculate the surface area of the solid formed by revolving the region bounded by f(x)=(1−x2)3,f ( x ) = \left( 1 - x ^ { 2 } \right) ^ { 3 },f(x)=(1−x2)3, the x-axis, x = -2, and x = 0 about the x-axis?

Let A denote the area enclosed by the equations y=x2+1y = x ^ { 2 } + 1y=x2+1 and y=−4x−2y = - 4 x - 2y=−4x−2 Then A is

Let yˉ\bar { y }yˉ​ denote the y-coordinate of the centroid of the region enclosed by y2=4x,y=0,y ^ { 2 } = 4 x , y = 0,y2=4x,y=0, x=1,x = 1,x=1, and x=1,x = 1,x=1, Then yˉ\bar { y }yˉ​ =

Let A denote the area enclosed by the equations y=3xy = \frac { 3 } { x }y=x3​ and y=4−xy = 4 - xy=4−x Then A is

Preparing for Calculus

Limits and Continuity

The Derivative

More About Derivatives

Applications of the Derivative

The Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

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Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

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Filters

Let $\bar { y }$ denote the y-coordinate of the centroid of the region enclosed by $y = x ^ { 2 }$ and $y = 1$ Then $\bar { y }$ =

The arc length of $y = ( x - 3 ) ^ { \frac { 3 } { 2 } } - 1 \text { for } x \in [ 3,6 ]$ is

Let $\bar { y }$ denote the y-coordinate of the centroid of the region enclosed by $y = x ^ { 2 }$ and $y = 2 x + 3$ Then $\bar { y }$ =

Let V be the volume of the solid generated by revolving the region enclosed by $y = \sqrt { x },$ the x-axis, $x = 1;$ about the y-axis. Then V is

Let V be the volume of the solid generated by revolving the region enclosed by $y = 2 | x | \text {, }$ the x-axis, $x = - 1 ;$ about the x-axis. Then V is

The arc length of $y = \frac { x ^ { 3 } } { 6 } + \frac { 1 } { 2 x } \text { for } x \in [ 2,5 ]$ is

The work done by a variable force $F ( x ) = ( 4 x - 1 ) ^ { 2 }$ Newtons that moves an object along a straight line in the direction of F from x = 2 meters to x = 4 meters, in Newton-meters, is

The arc length of $y = x ^ { \frac { 3 } { 2 } } + 9 \text { for } x \in [ 1,4 ]$ is

The arc length of $y = \frac { 4 } { 3 } ( x - 1 ) ^ { \frac { 3 } { 2 } } \text { for } x \in [ 2,4 ]$ is

Four particles having masses 2, 3, 3, and 4 kg are located on the xy-plane at $( - 1 , - 2 ) , ( 1,3 ) , ( 0,5 ) , ( 2,1 ),$ respectively. The x-coordinate of the center of mass of this system is located at $\bar { x } =$ ?

Let A denote the area enclosed by the equations $y = 3 - 2 | x |$ and $y = | x | \text {. }$ Then A is

A spring in equilibrium is 0.2 meters long and an external force of 0.5 Newtons stretches the spring to a length of 0.24 meters. The work done by the spring in stretching it from equilibrium to 0.28 meters, in Newton-meters, is

Which of these integrals can be used to calculate the surface area of the solid formed by revolving the region bounded by $f ( x ) = \left( 1 - x ^ { 2 } \right) ^ { 3 },$ the x-axis, x = -2, and x = 0 about the x-axis?

Let A denote the area enclosed by the equations $y = x ^ { 2 } + 1$ and $y = - 4 x - 2$ Then A is

Let $\bar { y }$ denote the y-coordinate of the centroid of the region enclosed by $y ^ { 2 } = 4 x , y = 0,$ $x = 1,$ and $x = 1,$ Then $\bar { y }$ =

Let A denote the area enclosed by the equations $y = \frac { 3 } { x }$ and $y = 4 - x$ Then A is