An aquarium 1 foot high, 1 foot wide, and 2 feet long is filled with water. For simplicity, take the density of water to be 60 lb/ft3 . Find the hydrostatic force in pound on one of the 1 foot by 2 foot sides of the aquarium.

A) 336 B) 30 C) 168 D) 240 E) 60 F)120 G)28 H)12 A) 336 B) 30 C) 168 D) 240 E) 60 F)120 G)28 H)12

Exam 6: Applications of Integration

A stone is thrown straight up from the top of a tower that is 80 ft tall with initial velocity 64 ft/s. What is the total distance traveled by the stone when it hits the ground?

(Short Answer)

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Question 21

The equation of a curve in parametric form is $x = 4 \cos 3 t , y = 4 \sin 3 t$ Find the arc length of the curve from $t = 0 \text { to } t = \frac { \pi } { 8 }$ .

(Short Answer)

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Question 22

Find the length of $y = \ln ( \sin x ) \text { for } \frac { \pi } { 6 } \leq x \leq \frac { \pi } { 3 }$

(Essay)

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Question 23

Find the area of the region bounded by the curves $x = 4 - y ^ { 2 }$ and $x = - 3 y$

(Multiple Choice)

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Question 24

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?

(Multiple Choice)

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Question 25

Find the average value of $f ( x ) = x \sqrt { 25 - x ^ { 2 } }$ on the interval $[ 0,5 ]$ . At how many points in the interval does $f ( x )$ have this value?

(Short Answer)

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Question 26

Find the volume of the solid obtained when the region above the x-axis, bounded by the x-axis and the curve $y = x - x ^ { 3 }$ , is rotated about the y-axis.

(Multiple Choice)

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Question 27

Let R be region bounded by the graph of $f ( x ) = x ^ { 2 } , g ( x ) = \frac { 1 } { x }$ , and the line y = 3.(a) Find the volume of the solid obtained by rotating R about the x-axis.(b) Find the volume of the solid obtained by rotating R about the y-axis.(c) Find the volume of the solid obtained by rotating R about the line y = 3.(d) Find the volume of the solid obtained by rotating R about the line x = 2.

(Essay)

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Question 28

Find the volume of the solid obtained by rotating the region bounded by the curve $y = \sqrt { 1 - x ^ { 2 } }$ and the x-axis about the line y = 2.

(Essay)

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Question 29

Find the volume of the solid obtained by rotating the region bounded by the curves $y = 3 - x ^ { 2 } \text { and } y = 2$ about the line y = 2.

(Short Answer)

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Question 30

Determine the centroid of the region bounded by the equation $y ^ { 2 } - 9 x = 0$ in the first quadrant between x = 1 and x = 4.

(Short Answer)

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Question 31

Find the length of the curve $y = x ^ { 3 } , 1 \leq x \leq 3$ using a graphing calculator to evaluate the integral

(Essay)

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Question 32

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 equilateral triangle with its base lying in R. Find the volume of the solid.

(Short Answer)

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Question 33

Find the area of the shaded region:

(Essay)

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Question 34

Find the volume of the solid obtained when the region bounded by the curve $y = \sin x , 0 \leq x \leq \pi$ and the x-axis is rotated about the x-axis.

(Multiple Choice)

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Question 35

Let $f ( x ) = \left\{ \begin{array} { l l } \frac { k \ln x } { x ^ { 4 } } & \text { if } x > 1 \\0 & \text { otherwise }\end{array} \right.$ (a) Find k so that f can serve as the probability density function of a random variable X.(b) Find $P ( X > e )$ .(c) Find the mean.

(Essay)

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Question 36

An aquarium 1 foot high, 1 foot wide, and 2 feet long is filled with water. For simplicity, take the density of water to be 60 lb/ft³. Find the hydrostatic force in pound on one of the 1 foot by 2 foot sides of the aquarium.

(Multiple Choice)

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Question 37

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 mean of the probability density function?

(Essay)

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Question 38

Find k so that the function $f ( x ) = \left\{ \begin{array} { l l } \frac { k } { x ( \ln x ) ^ { 3 } } & \text { if } x > e \\0 & \text { otherwise }\end{array} \right.$ can serve as the probability density function of a random variable X.

(Multiple Choice)

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Question 39

A hole of radius 6 cm is drilled through the center of a sphere of radius 10 cm. How much of the ball's volume is removed?

(Short Answer)

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Question 40

A stone is thrown straight up from the top of a tower that is 80 ft tall with initial velocity 64 ft/s. What is the total distance traveled by the stone when it hits the ground?

The equation of a curve in parametric form is $x = 4 \cos 3 t , y = 4 \sin 3 t$ Find the arc length of the curve from $t = 0 \text { to } t = \frac { \pi } { 8 }$ .

Find the length of $y = \ln ( \sin x ) \text { for } \frac { \pi } { 6 } \leq x \leq \frac { \pi } { 3 }$

Find the area of the region bounded by the curves $x = 4 - y ^ { 2 }$ and $x = - 3 y$

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 average value of $f ( x ) = x \sqrt { 25 - x ^ { 2 } }$ on the interval $[ 0,5 ]$ . At how many points in the interval does $f ( x )$ have this value?

Find the volume of the solid obtained when the region above the x-axis, bounded by the x-axis and the curve $y = x - x ^ { 3 }$ , is rotated about the y-axis.

Find the volume of the solid obtained by rotating the region bounded by the curve $y = \sqrt { 1 - x ^ { 2 } }$ and the x-axis about the line y = 2.

Find the volume of the solid obtained by rotating the region bounded by the curves $y = 3 - x ^ { 2 } \text { and } y = 2$ about the line y = 2.

Determine the centroid of the region bounded by the equation $y ^ { 2 } - 9 x = 0$ in the first quadrant between x = 1 and x = 4.

Find the length of the curve $y = x ^ { 3 } , 1 \leq x \leq 3$ using a graphing calculator to evaluate the integral

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 equilateral triangle with its base lying in R. Find the volume of the solid.

Find the area of the shaded region:

Find the volume of the solid obtained when the region bounded by the curve $y = \sin x , 0 \leq x \leq \pi$ and the x-axis is rotated about the x-axis.

Let $f ( x ) = \left\{ \begin{array} { l l } \frac { k \ln x } { x ^ { 4 } } & \text { if } x > 1 \\0 & \text { otherwise }\end{array} \right.$ (a) Find k so that f can serve as the probability density function of a random variable X.(b) Find $P ( X > e )$ .(c) Find the mean.

An aquarium 1 foot high, 1 foot wide, and 2 feet long is filled with water. For simplicity, take the density of water to be 60 lb/ft³. Find the hydrostatic force in pound on one of the 1 foot by 2 foot sides of the aquarium.

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 mean of the probability density function?

Find k so that the function $f ( x ) = \left\{ \begin{array} { l l } \frac { k } { x ( \ln x ) ^ { 3 } } & \text { if } x > e \\0 & \text { otherwise }\end{array} \right.$ can serve as the probability density function of a random variable X.

A hole of radius 6 cm is drilled through the center of a sphere of radius 10 cm. How much of the ball's volume is removed?

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Exam 6: Applications of Integration

A stone is thrown straight up from the top of a tower that is 80 ft tall with initial velocity 64 ft/s. What is the total distance traveled by the stone when it hits the ground?

The equation of a curve in parametric form is x=4cos⁡3t,y=4sin⁡3tx = 4 \cos 3 t , y = 4 \sin 3 tx=4cos3t,y=4sin3t Find the arc length of the curve from t=0 to t=π8t = 0 \text { to } t = \frac { \pi } { 8 }t=0 to t=8π​ .

Find the length of y=ln⁡(sin⁡x) for π6≤x≤π3y = \ln ( \sin x ) \text { for } \frac { \pi } { 6 } \leq x \leq \frac { \pi } { 3 }y=ln(sinx) for 6π​≤x≤3π​

Find the area of the region bounded by the curves x=4−y2x = 4 - y ^ { 2 }x=4−y2 and x=−3yx = - 3 yx=−3y

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 average value of f(x)=x25−x2f ( x ) = x \sqrt { 25 - x ^ { 2 } }f(x)=x25−x2​ on the interval [0,5][ 0,5 ][0,5] . At how many points in the interval does f(x)f ( x )f(x) have this value?

Find the volume of the solid obtained when the region above the x-axis, bounded by the x-axis and the curve y=x−x3y = x - x ^ { 3 }y=x−x3 , is rotated about the y-axis.

Find the volume of the solid obtained by rotating the region bounded by the curve y=1−x2y = \sqrt { 1 - x ^ { 2 } }y=1−x2​ and the x-axis about the line y = 2.

Find the volume of the solid obtained by rotating the region bounded by the curves y=3−x2 and y=2y = 3 - x ^ { 2 } \text { and } y = 2y=3−x2 and y=2 about the line y = 2.

Determine the centroid of the region bounded by the equation y2−9x=0y ^ { 2 } - 9 x = 0y2−9x=0 in the first quadrant between x = 1 and x = 4.

Find the length of the curve y=x3,1≤x≤3y = x ^ { 3 } , 1 \leq x \leq 3y=x3,1≤x≤3 using a graphing calculator to evaluate the integral

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 equilateral triangle with its base lying in R. Find the volume of the solid.

Find the area of the shaded region:

Find the volume of the solid obtained when the region bounded by the curve y=sin⁡x,0≤x≤πy = \sin x , 0 \leq x \leq \piy=sinx,0≤x≤π and the x-axis is rotated about the x-axis.

An aquarium 1 foot high, 1 foot wide, and 2 feet long is filled with water. For simplicity, take the density of water to be 60 lb/ft3 . Find the hydrostatic force in pound on one of the 1 foot by 2 foot sides of the aquarium.

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 mean of the probability density function?

A hole of radius 6 cm is drilled through the center of a sphere of radius 10 cm. How much of the ball's volume is removed?

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

The equation of a curve in parametric form is $x = 4 \cos 3 t , y = 4 \sin 3 t$ Find the arc length of the curve from $t = 0 \text { to } t = \frac { \pi } { 8 }$ .

Find the length of $y = \ln ( \sin x ) \text { for } \frac { \pi } { 6 } \leq x \leq \frac { \pi } { 3 }$

Find the area of the region bounded by the curves $x = 4 - y ^ { 2 }$ and $x = - 3 y$

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 average value of $f ( x ) = x \sqrt { 25 - x ^ { 2 } }$ on the interval $[ 0,5 ]$ . At how many points in the interval does $f ( x )$ have this value?

Find the volume of the solid obtained when the region above the x-axis, bounded by the x-axis and the curve $y = x - x ^ { 3 }$ , is rotated about the y-axis.

Find the volume of the solid obtained by rotating the region bounded by the curve $y = \sqrt { 1 - x ^ { 2 } }$ and the x-axis about the line y = 2.

Find the volume of the solid obtained by rotating the region bounded by the curves $y = 3 - x ^ { 2 } \text { and } y = 2$ about the line y = 2.

Determine the centroid of the region bounded by the equation $y ^ { 2 } - 9 x = 0$ in the first quadrant between x = 1 and x = 4.

Find the length of the curve $y = x ^ { 3 } , 1 \leq x \leq 3$ using a graphing calculator to evaluate the integral

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 equilateral triangle with its base lying in R. Find the volume of the solid.

Find the volume of the solid obtained when the region bounded by the curve $y = \sin x , 0 \leq x \leq \pi$ and the x-axis is rotated about the x-axis.

An aquarium 1 foot high, 1 foot wide, and 2 feet long is filled with water. For simplicity, take the density of water to be 60 lb/ft³. Find the hydrostatic force in pound on one of the 1 foot by 2 foot sides of the aquarium.

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 mean of the probability density function?