Question 1

Let $f(x)=\frac{2 c}{\left(1+x^{2}\right)}$ a) For what value of c is f a probability density function?
b) For that value of c, find P (-1 < X < 1).

Accepted Answer

The answer of Let $f(x)=\frac{2 c}{\left(1+x^{2}\right)}$ a) For what value...

Question 2

For any normal distribution, find $P(\mu-1.7 \sigma \leq X \leq \mu+1.7 \sigma)$ to two decimal places.

Accepted Answer

The answer of For any normal distribution, find \(P(\mu-1.7 \sigma...

Question 3

A cylindrical drum of diameter 2 ft and length 6 ft is lying on its side, submerged in water 16 ft deep. Find the force exerted by the water on one end of the drum to the nearest pound. (The weight density of water is 62.4 lb/ft3.)

Accepted Answer

Question 4

For a given commodity and pure competition, the number of units produced and the price per unit are determined as the coordinates of the point of intersection of the supply and demand curves. Given the demand curve $p=60-\frac{x}{30}$ and the supply curve $p=30+\frac{x}{20}$ ,
find the consumer surplus.

Accepted Answer

The answer of For a given commodity and pure competition,...

Question 5

Find the centroid of the region bounded by the graphs of the given equations. $y=x^{2 / 3}, \quad y=0, \quad x=125$

Accepted Answer

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

Question 6

If f (x) is the probability density function for the blood cholesterol level of men over the age of 40, where x is measured in milligrams per deciliter, express as an integral the probability that the cholesterol level of such a man lies between 195 and $250$ .

Accepted Answer

A)  $\int_{40}^{250} f(x) d x$ 
B)  $\int_{0}^{250} f(x) d x$ 
C)  $\int_{195}^{250} f(x) d x$ 
D)  $\int_{40}^{195} f(x) d x$ 
E)  $\int_{250}^{195} f(x) d x$ 
A)  $\int_{40}^{250} f(x) d x$ 
B)  $\int_{0}^{250} f(x) d x$ 
C)  $\int_{195}^{250} f(x) d x$ 
D)  $\int_{40}^{195} f(x) d x$ 
E)  $\int_{250}^{195} f(x) d x$

Question 7

Find the center of mass of the lamina of the region shown if the density of the circular lamina is five times that of the rectangular lamina.

Accepted Answer

The answer of Find the center of mass of the...

Question 8

Find the area of the surface obtained by revolving the given curve about the x-axis. $y=\frac{1}{2 \sqrt{10}} \sqrt{5 x^{2}-x^{4}}$ on [0, 1]

Accepted Answer

The answer of Find the area of the surface obtained...

Question 9

You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3.)

Accepted Answer

Question 10

Find the area of the region under the graph of f on [a, b]. $f(x)=x^{2}-2 x+2 ; \quad[-1,2]$

Accepted Answer

A)  -3 
B)  -6 
C)  6 
D)  3 
A)  -3 
B)  -6 
C)  6 
D)  3

Question 11

Find the centroid of the region bounded by the given curves. $y=14 x^{3}, 14 x+y=28, x=0$

Accepted Answer

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

Question 12

Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation 12
G) If the target weight is 500 g, what is the probability that the machine produces a box with less than $480$ g of cereal? Round your answer to four decimal places.

Accepted Answer

The answer of Boxes are labeled as containing 500 g...

Question 13

A vertical plate is submerged in water (the surface of the water coincides with the x-axis). Find the force exerted by the water on the plate. (The weight density of water is 62.4 lb/ft3.)   (ft)

Accepted Answer

A)  62.4 lb 
B)  873.6 lb 
C)  436.8 lb 
D)  124.8 lb 
A)  62.4 lb 
B)  873.6 lb 
C)  436.8 lb 
D)  124.8 lb

Question 14

A gate in an irrigation canal is constructed in the form of a trapezoid $6$ ft wide at the bottom, $12$ ft wide at the top, and 2 ft high. It is placed vertically in the canal, with the water extending to its top. Find the hydrostatic force on one side of the gate..

Accepted Answer

A)  $1015.8$ lb 
B)  $998.4$ lb 
C)  $1009.8$ lb 
D)  $973$ lb 
E)  None of these 
A)  $1015.8$ lb 
B)  $998.4$ lb 
C)  $1009.8$ lb 
D)  $973$ lb 
E)  None of these

Question 15

A swimming pool is 10 ft wide and 36 ft long and its bottom is an inclined plane, the shallow end having a depth of 1 ft and the deep end, 12 ft. If the pool is full of water, find the hydrostatic force on the shallow end. (Use the fact that water weighs 62.5 lb/ $\mathrm{ft}^{3}$ .)

Accepted Answer

The answer of A swimming pool is 10 ft wide...

Question 16

Find the centroid of the region bounded by the given curves. $y=9 x^{3}, 9 x+y=18, x=0$

Accepted Answer

A)  $\left(\frac{7461}{75}, \frac{261}{105}\right)$ 
B)  $\left(\frac{252}{75}, \frac{7452}{105}\right)$ 
C)  $\left(\frac{92}{105}, \frac{9}{75}\right)$ 
D)  $\left(\frac{261}{75}, \frac{7461}{105}\right)$ 
E)  None of these 
A)  $\left(\frac{7461}{75}, \frac{261}{105}\right)$ 
B)  $\left(\frac{252}{75}, \frac{7452}{105}\right)$ 
C)  $\left(\frac{92}{105}, \frac{9}{75}\right)$ 
D)  $\left(\frac{261}{75}, \frac{7461}{105}\right)$ 
E)  None of these

Question 17

Find the centroid of the region bounded by the graphs of the given equations. $y=15-x^{2}, \quad y=3-x$

Accepted Answer

A)  $\left(\frac{1}{2}, \frac{37}{5}\right)$ 
B)  $\left(\frac{1}{2}, \frac{5}{2}\right)$ 
C)  $\left(\frac{5}{2}, \frac{1}{2}\right)$ 
D)  $\left(\frac{37}{5}, \frac{1}{2}\right)$ 
A)  $\left(\frac{1}{2}, \frac{37}{5}\right)$ 
B)  $\left(\frac{1}{2}, \frac{5}{2}\right)$ 
C)  $\left(\frac{5}{2}, \frac{1}{2}\right)$ 
D)  $\left(\frac{37}{5}, \frac{1}{2}\right)$

Question 18

Find the exact coordinates of the centroid. $y=e^{7 x}, y=0, x=0, x=\frac{1}{7}$

Accepted Answer

The answer of Find the exact coordinates of the centroid....

Question 19

For the following exercise, (a) plot the graph of the function f, (b) write an integral giving the arc length of the graph of the function over the indicated interval, and (c) find the arc length of the curve accurate to two decimal places. $f(x)$ = x - $\sqrt{x}$ ; [0, 6]

Accepted Answer

The answer of For the following exercise, (a) plot the...

Question 20

Find the center of mass of the system comprising masses mk located at the points Pk in a coordinate plane. Assume that mass is measured in grams and distance is measured in centimeters.
m1 = 3, m2 = 4, m3 = 5
P1 (-3, 5), P2 (3, 4), P3 (-4, 1)

Accepted Answer

Let $f(x)=\frac{2 c}{\left(1+x^{2}\right)}$ a) For what value of c is f a probability density function? b) For that value of c, find P (-1 < X < 1).

For any normal distribution, find $P(\mu-1.7 \sigma \leq X \leq \mu+1.7 \sigma)$ to two decimal places.

A cylindrical drum of diameter 2 ft and length 6 ft is lying on its side, submerged in water 16 ft deep. Find the force exerted by the water on one end of the drum to the nearest pound. (The weight density of water is 62.4 lb/ft³.)

Find the centroid of the region bounded by the graphs of the given equations. $y=x^{2 / 3}, \quad y=0, \quad x=125$

If f (x) is the probability density function for the blood cholesterol level of men over the age of 40, where x is measured in milligrams per deciliter, express as an integral the probability that the cholesterol level of such a man lies between 195 and $250$ .

Find the center of mass of the lamina of the region shown if the density of the circular lamina is five times that of the rectangular lamina.

Find the area of the surface obtained by revolving the given curve about the x-axis. $y=\frac{1}{2 \sqrt{10}} \sqrt{5 x^{2}-x^{4}}$ on [0, 1]

You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft³.)

Find the area of the region under the graph of f on [a, b]. $f(x)=x^{2}-2 x+2 ; \quad[-1,2]$

Find the centroid of the region bounded by the given curves. $y=14 x^{3}, 14 x+y=28, x=0$

A vertical plate is submerged in water (the surface of the water coincides with the x-axis). Find the force exerted by the water on the plate. (The weight density of water is 62.4 lb/ft³.) (ft)

A gate in an irrigation canal is constructed in the form of a trapezoid $6$ ft wide at the bottom, $12$ ft wide at the top, and 2 ft high. It is placed vertically in the canal, with the water extending to its top. Find the hydrostatic force on one side of the gate..

A swimming pool is 10 ft wide and 36 ft long and its bottom is an inclined plane, the shallow end having a depth of 1 ft and the deep end, 12 ft. If the pool is full of water, find the hydrostatic force on the shallow end. (Use the fact that water weighs 62.5 lb/ $\mathrm{ft}^{3}$ .)

Find the centroid of the region bounded by the given curves. $y=9 x^{3}, 9 x+y=18, x=0$

Find the centroid of the region bounded by the graphs of the given equations. $y=15-x^{2}, \quad y=3-x$

Find the exact coordinates of the centroid. $y=e^{7 x}, y=0, x=0, x=\frac{1}{7}$

For the following exercise, (a) plot the graph of the function f, (b) write an integral giving the arc length of the graph of the function over the indicated interval, and (c) find the arc length of the curve accurate to two decimal places. $f(x)$ = x - $\sqrt{x}$ ; [0, 6]

Find the center of mass of the system comprising masses m_k located at the points P_k in a coordinate plane. Assume that mass is measured in grams and distance is measured in centimeters. m₁ = 3, m₂ = 4, m₃ = 5 P₁(-3, 5), P₂(3, 4), P₃(-4, 1)

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

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Exam 8: Further Applications of Integration

Let f(x)=2c(1+x2)f(x)=\frac{2 c}{\left(1+x^{2}\right)}f(x)=(1+x2)2c​ a) For what value of c is f a probability density function? b) For that value of c, find P (-1 < X < 1).

For any normal distribution, find P(μ−1.7σ≤X≤μ+1.7σ)P(\mu-1.7 \sigma \leq X \leq \mu+1.7 \sigma)P(μ−1.7σ≤X≤μ+1.7σ) to two decimal places.

A cylindrical drum of diameter 2 ft and length 6 ft is lying on its side, submerged in water 16 ft deep. Find the force exerted by the water on one end of the drum to the nearest pound. (The weight density of water is 62.4 lb/ft3.)

Find the centroid of the region bounded by the graphs of the given equations. y=x2/3,y=0,x=125y=x^{2 / 3}, \quad y=0, \quad x=125y=x2/3,y=0,x=125

If f (x) is the probability density function for the blood cholesterol level of men over the age of 40, where x is measured in milligrams per deciliter, express as an integral the probability that the cholesterol level of such a man lies between 195 and 250250250 .

Find the center of mass of the lamina of the region shown if the density of the circular lamina is five times that of the rectangular lamina.

Find the area of the surface obtained by revolving the given curve about the x-axis. y=12105x2−x4y=\frac{1}{2 \sqrt{10}} \sqrt{5 x^{2}-x^{4}}y=210​1​5x2−x4​ on [0, 1]

You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft3.)

Find the area of the region under the graph of f on [a, b]. f(x)=x2−2x+2;[−1,2]f(x)=x^{2}-2 x+2 ; \quad[-1,2]f(x)=x2−2x+2;[−1,2]

Find the centroid of the region bounded by the given curves. y=14x3,14x+y=28,x=0y=14 x^{3}, 14 x+y=28, x=0y=14x3,14x+y=28,x=0

A vertical plate is submerged in water (the surface of the water coincides with the x-axis). Find the force exerted by the water on the plate. (The weight density of water is 62.4 lb/ft3.) (ft)

A gate in an irrigation canal is constructed in the form of a trapezoid 666 ft wide at the bottom, 121212 ft wide at the top, and 2 ft high. It is placed vertically in the canal, with the water extending to its top. Find the hydrostatic force on one side of the gate..

A swimming pool is 10 ft wide and 36 ft long and its bottom is an inclined plane, the shallow end having a depth of 1 ft and the deep end, 12 ft. If the pool is full of water, find the hydrostatic force on the shallow end. (Use the fact that water weighs 62.5 lb/ ft3\mathrm{ft}^{3}ft3 .)

Find the centroid of the region bounded by the given curves. y=9x3,9x+y=18,x=0y=9 x^{3}, 9 x+y=18, x=0y=9x3,9x+y=18,x=0

Find the centroid of the region bounded by the graphs of the given equations. y=15−x2,y=3−xy=15-x^{2}, \quad y=3-xy=15−x2,y=3−x

Find the exact coordinates of the centroid. y=e7x,y=0,x=0,x=17y=e^{7 x}, y=0, x=0, x=\frac{1}{7}y=e7x,y=0,x=0,x=71​

For the following exercise, (a) plot the graph of the function f, (b) write an integral giving the arc length of the graph of the function over the indicated interval, and (c) find the arc length of the curve accurate to two decimal places. f(x)f(x)f(x) = x - x\sqrt{x}x​ ; [0, 6]

Find the center of mass of the system comprising masses mk located at the points Pk in a coordinate plane. Assume that mass is measured in grams and distance is measured in centimeters. m1 = 3, m2 = 4, m3 = 5 P1 (-3, 5), P2 (3, 4), P3 (-4, 1)

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Let $f(x)=\frac{2 c}{\left(1+x^{2}\right)}$ a) For what value of c is f a probability density function? b) For that value of c, find P (-1 < X < 1).

For any normal distribution, find $P(\mu-1.7 \sigma \leq X \leq \mu+1.7 \sigma)$ to two decimal places.

A cylindrical drum of diameter 2 ft and length 6 ft is lying on its side, submerged in water 16 ft deep. Find the force exerted by the water on one end of the drum to the nearest pound. (The weight density of water is 62.4 lb/ft³.)

Find the centroid of the region bounded by the graphs of the given equations. $y=x^{2 / 3}, \quad y=0, \quad x=125$

If f (x) is the probability density function for the blood cholesterol level of men over the age of 40, where x is measured in milligrams per deciliter, express as an integral the probability that the cholesterol level of such a man lies between 195 and $250$ .

Find the area of the surface obtained by revolving the given curve about the x-axis. $y=\frac{1}{2 \sqrt{10}} \sqrt{5 x^{2}-x^{4}}$ on [0, 1]

You are given the shape of the vertical ends of a trough that is completely filled with water. Find the force exerted by the water on one end of the trough. (The weight density of water is 62.4 lb/ft³.)

Find the area of the region under the graph of f on [a, b]. $f(x)=x^{2}-2 x+2 ; \quad[-1,2]$

Find the centroid of the region bounded by the given curves. $y=14 x^{3}, 14 x+y=28, x=0$

A vertical plate is submerged in water (the surface of the water coincides with the x-axis). Find the force exerted by the water on the plate. (The weight density of water is 62.4 lb/ft³.) (ft)

A gate in an irrigation canal is constructed in the form of a trapezoid $6$ ft wide at the bottom, $12$ ft wide at the top, and 2 ft high. It is placed vertically in the canal, with the water extending to its top. Find the hydrostatic force on one side of the gate..

A swimming pool is 10 ft wide and 36 ft long and its bottom is an inclined plane, the shallow end having a depth of 1 ft and the deep end, 12 ft. If the pool is full of water, find the hydrostatic force on the shallow end. (Use the fact that water weighs 62.5 lb/ $\mathrm{ft}^{3}$ .)

Find the centroid of the region bounded by the given curves. $y=9 x^{3}, 9 x+y=18, x=0$

Find the centroid of the region bounded by the graphs of the given equations. $y=15-x^{2}, \quad y=3-x$

Find the exact coordinates of the centroid. $y=e^{7 x}, y=0, x=0, x=\frac{1}{7}$

For the following exercise, (a) plot the graph of the function f, (b) write an integral giving the arc length of the graph of the function over the indicated interval, and (c) find the arc length of the curve accurate to two decimal places. $f(x)$ = x - $\sqrt{x}$ ; [0, 6]

Find the center of mass of the system comprising masses m_k located at the points P_k in a coordinate plane. Assume that mass is measured in grams and distance is measured in centimeters. m₁ = 3, m₂ = 4, m₃ = 5 P₁(-3, 5), P₂(3, 4), P₃(-4, 1)