Question 1

Find the average value of $f ( x ) = 6 - | x |$ on the interval $[ - 2,2 ]$ .

Accepted Answer

The answer of Find the average value of \(f (...

Question 2

Assume the daily consumption of electric power (in millions of kilowatt-hours) of a certain city
has the probability density $p ( x ) = \left\{ \begin{array} { l l } 
\frac { 1 } { 9 } x e ^ { - \frac { x } { 3 } } & \text { if } x \geq 0 \
0 & \text { if } x < 0
\end{array} \right.$ If the city's power plant has a daily capacity of 12 million kilowatt-hours, what is the probability that the available power supply will be inadequate on any given day?

Accepted Answer

The answer of Assume the daily consumption of electric power...

Question 3

Find the centroid of the region bounded by $y = \sin x , y = 0 \text { and } 0 \leq x \leq \pi$ .

Accepted Answer

The answer of Find the centroid of the region bounded...

Question 4

IQ scores are assumed to be normally distributed with a mean $\mu = 100$ and standard deviation $\sigma = 15$ . Use either Simpson's Rule or the Midpoint Rule to approximate the probability that a person selected at random from the general population will have an IQ score
(a) between 70 and 130.(b) over 130.

Accepted Answer

(a) About

Question 5

A force of 10 pounds is required to stretch a spring from its natural length of 8 inches to a length of 10 inches. How much work is done in stretching the spring to a length of 12 inches?

Accepted Answer

The answer of A force of 10 pounds is required...

Question 6

The following table shows the velocity of a car (in mi/hr) during the first five seconds of a race.$\begin{array} { | l | c | c | c | c | c | c | } 
\hline t ( s ) & 0 & 1 & 2 & 3 & 4 & 5 \
\hline v ( \mathrm { mi } / \mathrm { h } ) & 0 & 20 & 32 & 46 & 54 & 62 \
\hline
\end{array}$
Determine the average velocity of the car during this five-second interval.

Accepted Answer

Answer may

Question 7

Find the area of the shaded region:

Accepted Answer

The answer of Find the area of the shaded region:...

Question 8

Find the area of the region bounded by $y = x ( x - 1 ) ( x - 2 ) \text { and the } x - \text { axis. }$

Accepted Answer

The answer of Find the area of the region bounded...

Question 9

Let $f ( t ) = 0.2 e ^ { - 2 t } , t \geq 0$ be the probability density function of a random variable T, where t is the time that a customer spends in line at teller's window before being served. What is median of the probability density function?

Accepted Answer

The answer of Let \(f ( t ) = 0.2...

Question 10

The density of a rod 9 meters long is $\sqrt { x }$ kg/m at a distance of x meters from one end of the rod. Find the average density of the rod.

Accepted Answer

A) 6 
B)  $\frac { 4 } { 3 }$ 
C) 3 
D) 1 
E) 2
F) $\frac { 16 } { 3 }$ 
G)4
H) $\frac { 8 } { 3 }$ 
A) 6 
B)  $\frac { 4 } { 3 }$ 
C) 3 
D) 1 
E) 2
F) $\frac { 16 } { 3 }$ 
G)4
H) $\frac { 8 } { 3 }$

Question 11

Find the average value of the function whose graph is given below.

Accepted Answer

The answer of Find the average value of the function...

Question 12

The density function for the life of a certain type of battery is modeled by $p ( t ) = 0.2 e ^ { - 0.2 t } , t > 0$ and is measured in months.(a) Find the median life of the batteries.(b) Find the mean life of the batteries.(c) Sketch the graph of the density function showing the median and mean.

Accepted Answer

(a) 5 ln 2

Question 13

Find the average value of the function $f ( x ) = \frac { x } { \sqrt { x ^ { 2 } + 9 } }$ $0 \leq x \leq 4$ .

Accepted Answer

A) 0 
B)  $\frac { 2 } { 5 }$ 
C) 10 
D)  $\frac { 1 } { 5 }$ 
E)  $\frac { 1 } { 2 }$ 
F)5
G)4

H) $\frac { 4 } { 5 }$ 
A) 0 
B)  $\frac { 2 } { 5 }$ 
C) 10 
D)  $\frac { 1 } { 5 }$ 
E)  $\frac { 1 } { 2 }$ 
F)5
G)4

H) $\frac { 4 } { 5 }$

Question 14

Find the center of mass of the system $m _ { 1 } = 5 , P _ { 1 } ( - 3,1 ) , m _ { 2 } = 9$ $P _ { 2 } = ( - 1 , - 1 ) , m _ { 3 } = 6$ $P _ { 3 } = ( 1,1 ) , m _ { 4 } = 8 , P _ { 4 } ( 3 , - 2 )$

Accepted Answer

A)  $\left( \frac { 3 } { 14 } , - \frac { 1 } { 2 } \right)$ 
B)  $\left( - \frac { 1 } { 2 } , \frac { 3 } { 14 } \right)$ 
C)  $\left( \frac { 3 } { 14 } , \frac { 1 } { 2 } \right)$ 
D)  $\left( - \frac { 3 } { 14 } , \frac { 1 } { 2 } \right)$ 
E)  $( 6 , - 14 )$ 
F) $( - 14,6 )$ 
G) $( 14 , - 6 )$ 
H) $( - 14 , - 6 )$ 
A)  $\left( \frac { 3 } { 14 } , - \frac { 1 } { 2 } \right)$ 
B)  $\left( - \frac { 1 } { 2 } , \frac { 3 } { 14 } \right)$ 
C)  $\left( \frac { 3 } { 14 } , \frac { 1 } { 2 } \right)$ 
D)  $\left( - \frac { 3 } { 14 } , \frac { 1 } { 2 } \right)$ 
E)  $( 6 , - 14 )$ 
F) $( - 14,6 )$ 
G) $( 14 , - 6 )$ 
H) $( - 14 , - 6 )$

Question 15

Find the average value of the function $f ( x ) = \sqrt { 4 - x ^ { 2 } }$ on the interval $[ - 2,2 ]$ .

Accepted Answer

A)  $\frac { \pi } { 2 }$ 
B)  $\pi$ 
C) 8 $\pi$ 
D) 3 $\pi$ 
E) 2 $\pi$ 
F)4 $\pi$ 
G)16 $\pi$ 
H)6 $\pi$ 
A)  $\frac { \pi } { 2 }$ 
B)  $\pi$ 
C) 8 $\pi$ 
D) 3 $\pi$ 
E) 2 $\pi$ 
F)4 $\pi$ 
G)16 $\pi$ 
H)6 $\pi$

Question 16

The demand function for a certain commodity is $p ( x ) = \frac { 1800 } { ( x + 5 ) ^ { 2 } }$ . Find the consumer surplus when the selling price is $18.

Accepted Answer

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

Question 17

Find the area of the region bounded by $x = e ^ { t } \cos t , y = e ^ { t } \sin t , 0 \leq t \leq \pi$ , and the x-axis.

Accepted Answer

A)  $\frac { 1 } { 4 } e ^ { 2 x }$ 
B)  $\frac { 1 } { 2 } e ^ { 2 x }$ 
C)  $\frac { 1 } { 4 } \left( e ^ { 2 x } + 1 \right)$ 
D)  $\frac { 1 } { 4 } e ^ { x }$ 
E)  $\frac { 1 } { 4 } \left( e ^ { 2 x } - 1 \right)$ 
F) $\frac { 1 } { 2 } \left( e ^ { 2 x } - 1 \right)$ 
G) $\frac { 1 } { 2 } \left( e ^ { x } - 1 \right)$ 
H) $e ^ { 2 x } - 1$ 
A)  $\frac { 1 } { 4 } e ^ { 2 x }$ 
B)  $\frac { 1 } { 2 } e ^ { 2 x }$ 
C)  $\frac { 1 } { 4 } \left( e ^ { 2 x } + 1 \right)$ 
D)  $\frac { 1 } { 4 } e ^ { x }$ 
E)  $\frac { 1 } { 4 } \left( e ^ { 2 x } - 1 \right)$ 
F) $\frac { 1 } { 2 } \left( e ^ { 2 x } - 1 \right)$ 
G) $\frac { 1 } { 2 } \left( e ^ { x } - 1 \right)$ 
H) $e ^ { 2 x } - 1$

Question 18

Find the length of the curve $x = \frac { 1 } { 3 } \left( y ^ { 2 } + 2 \right) ^ { \frac { 3 } { 2 } } , 1 \leq y \leq 2$

Accepted Answer

A) 0 
B)  $\frac { 7 } { 3 }$ 
C) 2 
D)  $\frac { 14 } { 3 }$ 
E) 1
F) $\frac { 5 } { 3 }$ 
G)4

H) $\frac { 10 } { 3 }$ 
A) 0 
B)  $\frac { 7 } { 3 }$ 
C) 2 
D)  $\frac { 14 } { 3 }$ 
E) 1
F) $\frac { 5 } { 3 }$ 
G)4

H) $\frac { 10 } { 3 }$

Question 19

Using the help of a graphing calculator, find the area of the region bounded by the curves $y = x ( x - 1 )$ and $y = \sin ( x )$ .

Accepted Answer

The answer of Using the help of a graphing calculator,...

Question 20

Find the centroid of the region bounded by $y = \cos x , y = 0 \text { and } 0 \leq x \leq \frac { \pi } { 2 }$ .

Accepted Answer

The answer of Find the centroid of the region bounded...

Find the average value of $f ( x ) = 6 - | x |$ on the interval $[ - 2,2 ]$ .

Find the centroid of the region bounded by $y = \sin x , y = 0 \text { and } 0 \leq x \leq \pi$ .

A force of 10 pounds is required to stretch a spring from its natural length of 8 inches to a length of 10 inches. How much work is done in stretching the spring to a length of 12 inches?

The following table shows the velocity of a car (in mi/hr) during the first five seconds of a race. t(s) 0 1 2 3 4 5 v(/) 0 20 32 46 54 62 Determine the average velocity of the car during this five-second interval.

Find the area of the shaded region:

Find the area of the region bounded by $y = x ( x - 1 ) ( x - 2 ) \text { and the } x - \text { axis. }$

Let $f ( t ) = 0.2 e ^ { - 2 t } , t \geq 0$ be the probability density function of a random variable T, where t is the time that a customer spends in line at teller's window before being served. What is median of the probability density function?

The density of a rod 9 meters long is $\sqrt { x }$ kg/m at a distance of x meters from one end of the rod. Find the average density of the rod.

Find the average value of the function whose graph is given below.

The density function for the life of a certain type of battery is modeled by $p ( t ) = 0.2 e ^ { - 0.2 t } , t > 0$ and is measured in months.(a) Find the median life of the batteries.(b) Find the mean life of the batteries.(c) Sketch the graph of the density function showing the median and mean.

Find the average value of the function $f ( x ) = \frac { x } { \sqrt { x ^ { 2 } + 9 } }$ $0 \leq x \leq 4$ .

Find the center of mass of the system $m _ { 1 } = 5 , P _ { 1 } ( - 3,1 ) , m _ { 2 } = 9$ $P _ { 2 } = ( - 1 , - 1 ) , m _ { 3 } = 6$ $P _ { 3 } = ( 1,1 ) , m _ { 4 } = 8 , P _ { 4 } ( 3 , - 2 )$

Find the average value of the function $f ( x ) = \sqrt { 4 - x ^ { 2 } }$ on the interval $[ - 2,2 ]$ .

The demand function for a certain commodity is $p ( x ) = \frac { 1800 } { ( x + 5 ) ^ { 2 } }$ . Find the consumer surplus when the selling price is $18.

Find the area of the region bounded by $x = e ^ { t } \cos t , y = e ^ { t } \sin t , 0 \leq t \leq \pi$ , and the x-axis.

Find the length of the curve $x = \frac { 1 } { 3 } \left( y ^ { 2 } + 2 \right) ^ { \frac { 3 } { 2 } } , 1 \leq y \leq 2$

Using the help of a graphing calculator, find the area of the region bounded by the curves $y = x ( x - 1 )$ and $y = \sin ( x )$ .

Find the centroid of the region bounded by $y = \cos x , y = 0 \text { and } 0 \leq x \leq \frac { \pi } { 2 }$ .

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Exam 6: Applications of Integration

Find the average value of f(x)=6−∣x∣f ( x ) = 6 - | x |f(x)=6−∣x∣ on the interval [−2,2][ - 2,2 ][−2,2] .

Find the centroid of the region bounded by y=sin⁡x,y=0 and 0≤x≤πy = \sin x , y = 0 \text { and } 0 \leq x \leq \piy=sinx,y=0 and 0≤x≤π .

A force of 10 pounds is required to stretch a spring from its natural length of 8 inches to a length of 10 inches. How much work is done in stretching the spring to a length of 12 inches?

The following table shows the velocity of a car (in mi/hr) during the first five seconds of a race. t(s) 0 1 2 3 4 5 v(/) 0 20 32 46 54 62 Determine the average velocity of the car during this five-second interval.

Find the area of the shaded region:

Find the area of the region bounded by y=x(x−1)(x−2) and the x− axis. y = x ( x - 1 ) ( x - 2 ) \text { and the } x - \text { axis. }y=x(x−1)(x−2) and the x− axis.

Let f(t)=0.2e−2t,t≥0f ( t ) = 0.2 e ^ { - 2 t } , t \geq 0f(t)=0.2e−2t,t≥0 be the probability density function of a random variable T, where t is the time that a customer spends in line at teller's window before being served. What is median of the probability density function?

The density of a rod 9 meters long is x\sqrt { x }x​ kg/m at a distance of x meters from one end of the rod. Find the average density of the rod.

Find the average value of the function whose graph is given below.

Find the average value of the function f(x)=xx2+9f ( x ) = \frac { x } { \sqrt { x ^ { 2 } + 9 } }f(x)=x2+9​x​ 0≤x≤40 \leq x \leq 40≤x≤4 .

Find the average value of the function f(x)=4−x2f ( x ) = \sqrt { 4 - x ^ { 2 } }f(x)=4−x2​ on the interval [−2,2][ - 2,2 ][−2,2] .

The demand function for a certain commodity is p(x)=1800(x+5)2p ( x ) = \frac { 1800 } { ( x + 5 ) ^ { 2 } }p(x)=(x+5)21800​ . Find the consumer surplus when the selling price is $18.

Find the area of the region bounded by x=etcos⁡t,y=etsin⁡t,0≤t≤πx = e ^ { t } \cos t , y = e ^ { t } \sin t , 0 \leq t \leq \pix=etcost,y=etsint,0≤t≤π , and the x-axis.

Find the length of the curve x=13(y2+2)32,1≤y≤2x = \frac { 1 } { 3 } \left( y ^ { 2 } + 2 \right) ^ { \frac { 3 } { 2 } } , 1 \leq y \leq 2x=31​(y2+2)23​,1≤y≤2

Using the help of a graphing calculator, find the area of the region bounded by the curves y=x(x−1)y = x ( x - 1 )y=x(x−1) and y=sin⁡(x)y = \sin ( x )y=sin(x) .

Find the centroid of the region bounded by y=cos⁡x,y=0 and 0≤x≤π2y = \cos x , y = 0 \text { and } 0 \leq x \leq \frac { \pi } { 2 }y=cosx,y=0 and 0≤x≤2π​ .

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Differential Equations

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Find the average value of $f ( x ) = 6 - | x |$ on the interval $[ - 2,2 ]$ .

Find the centroid of the region bounded by $y = \sin x , y = 0 \text { and } 0 \leq x \leq \pi$ .

Find the area of the region bounded by $y = x ( x - 1 ) ( x - 2 ) \text { and the } x - \text { axis. }$

Let $f ( t ) = 0.2 e ^ { - 2 t } , t \geq 0$ be the probability density function of a random variable T, where t is the time that a customer spends in line at teller's window before being served. What is median of the probability density function?

The density of a rod 9 meters long is $\sqrt { x }$ kg/m at a distance of x meters from one end of the rod. Find the average density of the rod.

Find the average value of the function $f ( x ) = \frac { x } { \sqrt { x ^ { 2 } + 9 } }$ $0 \leq x \leq 4$ .

Find the average value of the function $f ( x ) = \sqrt { 4 - x ^ { 2 } }$ on the interval $[ - 2,2 ]$ .

The demand function for a certain commodity is $p ( x ) = \frac { 1800 } { ( x + 5 ) ^ { 2 } }$ . Find the consumer surplus when the selling price is $18.

Find the area of the region bounded by $x = e ^ { t } \cos t , y = e ^ { t } \sin t , 0 \leq t \leq \pi$ , and the x-axis.

Find the length of the curve $x = \frac { 1 } { 3 } \left( y ^ { 2 } + 2 \right) ^ { \frac { 3 } { 2 } } , 1 \leq y \leq 2$

Using the help of a graphing calculator, find the area of the region bounded by the curves $y = x ( x - 1 )$ and $y = \sin ( x )$ .

Find the centroid of the region bounded by $y = \cos x , y = 0 \text { and } 0 \leq x \leq \frac { \pi } { 2 }$ .