Question 1

The base of a certain solid is a plane region R enclosed by the x-axis and the curve $y = 1 - x ^ { 2 }$ . Each cross-section of the solid perpendicular to the y-axis is an isosceles triangle of height 4 with its base lying in R. Find the volume of the solid.

Accepted Answer

The answer of The base of a certain solid is...

Question 2

A manufacture has been selling 1000 ceiling fans at $60 each. A market survey indicates that for every $10 that price is reduced, the number of sets sold will increase by 100. Find the demand function and calculate the consumer surplus when the selling price is set at $50.

Accepted Answer

The answer of A manufacture has been selling 1000 ceiling...

Question 3

A supply function is given by $p _ { 3 } ( x ) = 5 + \frac { 1 } { 10 } x$ , where x is the number of units produced. Find the producer surplus when the selling price is $15.

Accepted Answer

A) 100 
B) 500 
C) 250 
D) 1000 
E) 800
F)400
G)200
H)2000

Question 4

(a) Show that $$f ( x ) = \left\{ \begin{array} { l l } 
4 x + 12 x ^ { 2 } & \text { if } 0 \leq x \leq \frac { 1 } { 2 } \
0 & \text { otherwise }
\end{array} \right.$$ is the probability density function of a random variable.(b) What is the mean for this distribution?
(c) Calculate the median of $f$

Accepted Answer

The answer of (a) Show that \[f ( x )...

Question 5

Let $f ( x ) = 6 x ( 1 - x ) , 0 < x < 1$ be the probability density function of a random variable X. Find the mean of the probability density function $f$ .

Accepted Answer

A)  $\frac { 1 } { 3 }$ 
B)  $\frac { 1 } { 6 }$ 
C)  $\frac { 1 } { 2 }$ 
D) 1 
E) 3
F)6
G)2
H)0

Question 6

Find the volume of the solid obtained by rotating the region bounded by xy = 1, y = 1, y = 2 and the y-axis about the x-axis.

Accepted Answer

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

Question 7

Find k so that the function $f ( x ) = \left\{ \begin{array} { l l } 
k x ( 1 - x ) & \text { if } 0 < x < 1 \
0 & \text { otherwise }
\end{array} \right.$ can serve as the probability density function of a random variable X.

Accepted Answer

A)  $\frac { 1 } { 3 }$ 
B)  $\frac { 1 } { 6 }$ 
C)  $\frac { 1 } { 2 }$ 
D) 1 
E) 3
F)6
G)2
H)0

Question 8

Find the x-coordinate x at the centroid of the region bounded by the x-axis and the lines $y = 3 x , \quad x = 2$

Accepted Answer

A)  $\frac { 4 } { 3 }$ 
B)  $\frac { 7 } { 6 }$ 
C)  $\frac { 10 } { 7 }$ 
D)  $\frac { 5 } { 3 }$ 
E)  $\frac { 11 } { 7 }$ 
F) $\frac { 11 } { 6 }$ 
G) $\frac { 5 } { 4 }$ 
H) $\frac { 7 } { 4 }$

Question 9

Find the volume of the solid obtained when the region bounded by the line $y = 2 x$ , the line $x = 3$ , and the x-axis is rotated about the y-axis.

Accepted Answer

A)  $36 \pi$ 
B)  $42 \pi$ 
C)  $16 \pi$ 
D)  $18 \pi$ 
E)  $24 \pi$ 
F) $30 \pi$ 
G) $48 \pi$ 
H) $27 \pi$

Question 10

The volume of the solid obtained by rotating the plane region enclosed by $y = x ^ { 4 } , y = 1 , \text { and } x = 0$ about the y-axis is

Accepted Answer

A)  $\pi$ 
B)  $\frac { 2 \pi } { 3 }$ 
C)  $\frac { \pi } { 2 }$ 
D) 0 
E)  $\frac { \pi } { 3 }$ 
F) $\frac { 4 \pi } { 3 }$ 
G) $\frac { \pi } { 6 }$ 
H)1

Question 11

Find the y-coordinate of the centroid of the region bounded by the curves $y = x ^ { 2 } , y = 1$

Accepted Answer

A) 0.85 
B) 0.70 
C) 0.60 
D) 0.50 
E) 0.75
F)0.55
G)0.80
H)0.65

Question 12

Find the distance traveled by a particle with position $( x , y ) = \left( \cos ^ { 2 } t , \cos t \right)$ as t varies in the time interval $[ 0,4 \pi ]$ . Compare with the length of the curve.

Accepted Answer

The answer of Find the distance traveled by a particle...

Question 13

Let $f ( x ) = 6 x ( 1 - x ) , 0 < x < 1$ be the probability density function of a random variable X. Find the median of the probability density function $f$ .

Accepted Answer

A)  $\frac { 1 } { 3 }$ 
B)  $\frac { 1 } { 6 }$ 
C)  $\frac { 1 } { 2 }$ 
D) 1 
E) 3
F)6
G)2
H)0

Question 14

A particle is moving in a straight line and its velocity is given by $v ( t ) = 3 t ^ { 2 } - 12 t + 9,$ where t is measure in seconds and v in meters per second. Find the distance traveled by the particle during the time interval $[ 0,5 ]$ .

Accepted Answer

The answer of A particle is moving in a straight...

Question 15

The demand function for a certain commodity is $p = 5 - \frac { x } { 10 }$ Find the consumer's surplus when the sales level is 30. Illustrate by drawing the demand curve and identifying the consumer's surplus as an area.

Accepted Answer

The answer of The demand function for a certain commodity...

Question 16

The volume of the solid obtained by rotating the plane region enclosed by $y = \frac { \sin ( 2 x ) } { x } , x = \frac { \pi } { 2 } ,$ the y-axis and the x-axis about the y-axis is

Accepted Answer

A)  $2 \pi$ 
B)  $\pi$ 
C) 2 
D) 1 
E)  $4 \pi$ 
F) $\frac { \pi } { 2 }$ 
G)4
H) $\frac { 1 } { 12 }$

Question 17

Set up, but do not evaluate, an integral for the length of $y = x ^ { 4 } - x ^ { 2 } , - 1 \leq x \leq 2$

Accepted Answer

The answer of Set up, but do not evaluate, an...

Question 18

The region R is given by the shaded area in the figure below:  (a) Find the area of the shaded region R.(b) Find the volume of the solid obtained by rotating R about
(i) the x-axis. (ii) the y-axis. (iii) the line x = 2. (iv) the line y = 4

Accepted Answer

The answer of The region R is given by the...

Question 19

If $x = \cos 2 t , y = \sin ^ { 2 } t \text { and } ( x , y )$ represents the position of a particle, find the distance the particle travels as t moves from 0 to $\frac { \pi } { 2 } .$

Accepted Answer

The answer of If \(x = \cos 2 t ,...

Question 20

A hemispherical tank with radius 8 feet is filled with water to a depth of 6 feet. Find the work required to empty the tank by pumping the water to the top of the tank.

Accepted Answer

The answer of A hemispherical tank with radius 8 feet...

Exam 6: Applications of Integration

The base of a certain solid is a plane region R enclosed by the x-axis and the curve $y = 1 - x ^ { 2 }$ . Each cross-section of the solid perpendicular to the y-axis is an isosceles triangle of height 4 with its base lying in R. Find the volume of the solid.

A manufacture has been selling 1000 ceiling fans at $60 each. A market survey indicates that for every $10 that price is reduced, the number of sets sold will increase by 100. Find the demand function and calculate the consumer surplus when the selling price is set at $50.

A supply function is given by $p _ { 3 } ( x ) = 5 + \frac { 1 } { 10 } x$ , where x is the number of units produced. Find the producer surplus when the selling price is $15.

(a) Show that $f ( x ) = \left\{ \begin{array} { l l } 4 x + 12 x ^ { 2 } & \text { if } 0 \leq x \leq \frac { 1 } { 2 } \\0 & \text { otherwise }\end{array} \right.$ is the probability density function of a random variable.(b) What is the mean for this distribution? (c) Calculate the median of $f$

Let $f ( x ) = 6 x ( 1 - x ) , 0 < x < 1$ be the probability density function of a random variable X. Find the mean of the probability density function $f$ .

Find the volume of the solid obtained by rotating the region bounded by xy = 1, y = 1, y = 2 and the y-axis about the x-axis.

Find k so that the function $f ( x ) = \left\{ \begin{array} { l l } k x ( 1 - x ) & \text { if } 0 < x < 1 \\0 & \text { otherwise }\end{array} \right.$ can serve as the probability density function of a random variable X.

Find the x-coordinate x at the centroid of the region bounded by the x-axis and the lines $y = 3 x , \quad x = 2$

Find the volume of the solid obtained when the region bounded by the line $y = 2 x$ , the line $x = 3$ , and the x-axis is rotated about the y-axis.

The volume of the solid obtained by rotating the plane region enclosed by $y = x ^ { 4 } , y = 1 , \text { and } x = 0$ about the y-axis is

Find the y-coordinate of the centroid of the region bounded by the curves $y = x ^ { 2 } , y = 1$

Find the distance traveled by a particle with position $( x , y ) = \left( \cos ^ { 2 } t , \cos t \right)$ as t varies in the time interval $[ 0,4 \pi ]$ . Compare with the length of the curve.

Let $f ( x ) = 6 x ( 1 - x ) , 0 < x < 1$ be the probability density function of a random variable X. Find the median of the probability density function $f$ .

A particle is moving in a straight line and its velocity is given by $v ( t ) = 3 t ^ { 2 } - 12 t + 9,$ where t is measure in seconds and v in meters per second. Find the distance traveled by the particle during the time interval $[ 0,5 ]$ .

The demand function for a certain commodity is $p = 5 - \frac { x } { 10 }$ Find the consumer's surplus when the sales level is 30. Illustrate by drawing the demand curve and identifying the consumer's surplus as an area.

The volume of the solid obtained by rotating the plane region enclosed by $y = \frac { \sin ( 2 x ) } { x } , x = \frac { \pi } { 2 } ,$ the y-axis and the x-axis about the y-axis is

Set up, but do not evaluate, an integral for the length of $y = x ^ { 4 } - x ^ { 2 } , - 1 \leq x \leq 2$

The region R is given by the shaded area in the figure below: (a) Find the area of the shaded region R.(b) Find the volume of the solid obtained by rotating R about (i) the x-axis. (ii) the y-axis. (iii) the line x = 2. (iv) the line y = 4

If $x = \cos 2 t , y = \sin ^ { 2 } t \text { and } ( x , y )$ represents the position of a particle, find the distance the particle travels as t moves from 0 to $\frac { \pi } { 2 } .$

A hemispherical tank with radius 8 feet is filled with water to a depth of 6 feet. Find the work required to empty the tank by pumping the water to the top of the tank.

Exam 6: Applications of Integration

The base of a certain solid is a plane region R enclosed by the x-axis and the curve y=1−x2y = 1 - x ^ { 2 }y=1−x2 . Each cross-section of the solid perpendicular to the y-axis is an isosceles triangle of height 4 with its base lying in R. Find the volume of the solid.

A manufacture has been selling 1000 ceiling fans at $60 each. A market survey indicates that for every $10 that price is reduced, the number of sets sold will increase by 100. Find the demand function and calculate the consumer surplus when the selling price is set at $50.

A supply function is given by p3(x) =5+110xp _ { 3 } ( x ) = 5 + \frac { 1 } { 10 } xp3​(x) =5+101​x , where x is the number of units produced. Find the producer surplus when the selling price is $15.

Let f(x) =6x(1−x) ,0<x<1f ( x ) = 6 x ( 1 - x ) , 0 < x < 1f(x) =6x(1−x) ,0<x<1 be the probability density function of a random variable X. Find the mean of the probability density function fff .

Find the volume of the solid obtained by rotating the region bounded by xy = 1, y = 1, y = 2 and the y-axis about the x-axis.

Find the x-coordinate x at the centroid of the region bounded by the x-axis and the lines y=3x,x=2y = 3 x , \quad x = 2y=3x,x=2

Find the volume of the solid obtained when the region bounded by the line y=2xy = 2 xy=2x , the line x=3x = 3x=3 , and the x-axis is rotated about the y-axis.

The volume of the solid obtained by rotating the plane region enclosed by y=x4,y=1, and x=0y = x ^ { 4 } , y = 1 , \text { and } x = 0y=x4,y=1, and x=0 about the y-axis is

Find the y-coordinate of the centroid of the region bounded by the curves y=x2,y=1y = x ^ { 2 } , y = 1y=x2,y=1

Find the distance traveled by a particle with position (x,y)=(cos⁡2t,cos⁡t)( x , y ) = \left( \cos ^ { 2 } t , \cos t \right)(x,y)=(cos2t,cost) as t varies in the time interval [0,4π][ 0,4 \pi ][0,4π] . Compare with the length of the curve.

Let f(x) =6x(1−x) ,0<x<1f ( x ) = 6 x ( 1 - x ) , 0 < x < 1f(x) =6x(1−x) ,0<x<1 be the probability density function of a random variable X. Find the median of the probability density function fff .

A particle is moving in a straight line and its velocity is given by v(t)=3t2−12t+9,v ( t ) = 3 t ^ { 2 } - 12 t + 9,v(t)=3t2−12t+9, where t is measure in seconds and v in meters per second. Find the distance traveled by the particle during the time interval [0,5][ 0,5 ][0,5] .

The demand function for a certain commodity is p=5−x10p = 5 - \frac { x } { 10 }p=5−10x​ Find the consumer's surplus when the sales level is 30. Illustrate by drawing the demand curve and identifying the consumer's surplus as an area.

The volume of the solid obtained by rotating the plane region enclosed by y=sin⁡(2x) x,x=π2,y = \frac { \sin ( 2 x ) } { x } , x = \frac { \pi } { 2 } ,y=xsin(2x) ​,x=2π​, the y-axis and the x-axis about the y-axis is

Set up, but do not evaluate, an integral for the length of y=x4−x2,−1≤x≤2y = x ^ { 4 } - x ^ { 2 } , - 1 \leq x \leq 2y=x4−x2,−1≤x≤2

The region R is given by the shaded area in the figure below: (a) Find the area of the shaded region R.(b) Find the volume of the solid obtained by rotating R about (i) the x-axis. (ii) the y-axis. (iii) the line x = 2. (iv) the line y = 4

If x=cos⁡2t,y=sin⁡2t and (x,y)x = \cos 2 t , y = \sin ^ { 2 } t \text { and } ( x , y )x=cos2t,y=sin2t and (x,y) represents the position of a particle, find the distance the particle travels as t moves from 0 to π2.\frac { \pi } { 2 } .2π​.

A hemispherical tank with radius 8 feet is filled with water to a depth of 6 feet. Find the work required to empty the tank by pumping the water to the top of the tank.

The base of a certain solid is a plane region R enclosed by the x-axis and the curve $y = 1 - x ^ { 2 }$ . Each cross-section of the solid perpendicular to the y-axis is an isosceles triangle of height 4 with its base lying in R. Find the volume of the solid.

A supply function is given by $p _ { 3 } ( x ) = 5 + \frac { 1 } { 10 } x$ , where x is the number of units produced. Find the producer surplus when the selling price is $15.

Let $f ( x ) = 6 x ( 1 - x ) , 0 < x < 1$ be the probability density function of a random variable X. Find the mean of the probability density function $f$ .

Find the x-coordinate x at the centroid of the region bounded by the x-axis and the lines $y = 3 x , \quad x = 2$

Find the volume of the solid obtained when the region bounded by the line $y = 2 x$ , the line $x = 3$ , and the x-axis is rotated about the y-axis.

The volume of the solid obtained by rotating the plane region enclosed by $y = x ^ { 4 } , y = 1 , \text { and } x = 0$ about the y-axis is

Find the y-coordinate of the centroid of the region bounded by the curves $y = x ^ { 2 } , y = 1$

Find the distance traveled by a particle with position $( x , y ) = \left( \cos ^ { 2 } t , \cos t \right)$ as t varies in the time interval $[ 0,4 \pi ]$ . Compare with the length of the curve.

Let $f ( x ) = 6 x ( 1 - x ) , 0 < x < 1$ be the probability density function of a random variable X. Find the median of the probability density function $f$ .

A particle is moving in a straight line and its velocity is given by $v ( t ) = 3 t ^ { 2 } - 12 t + 9,$ where t is measure in seconds and v in meters per second. Find the distance traveled by the particle during the time interval $[ 0,5 ]$ .

The demand function for a certain commodity is $p = 5 - \frac { x } { 10 }$ Find the consumer's surplus when the sales level is 30. Illustrate by drawing the demand curve and identifying the consumer's surplus as an area.

The volume of the solid obtained by rotating the plane region enclosed by $y = \frac { \sin ( 2 x ) } { x } , x = \frac { \pi } { 2 } ,$ the y-axis and the x-axis about the y-axis is

Set up, but do not evaluate, an integral for the length of $y = x ^ { 4 } - x ^ { 2 } , - 1 \leq x \leq 2$

If $x = \cos 2 t , y = \sin ^ { 2 } t \text { and } ( x , y )$ represents the position of a particle, find the distance the particle travels as t moves from 0 to $\frac { \pi } { 2 } .$