Find the exact area of the surface obtained by rotating the given curve about the x-axis. :
a) y = 1/x where 1 ≤ x ≤ 4
b) y = (x2 +1)1/2 where 0 ≤ x ≤ 6
Answers
a) To find the area of the surface obtained by rotating y = 1/x about the x-axis, we use the formula:
A = 2π∫[a,b] y ds
where ds = √(1 + (dy/dx)^2) dx is the infinitesimal arc length, and a and b are the limits of integration.
Here, y = 1/x, so dy/dx = -1/x^2. Therefore,
ds = √(1 + (-1/x^2)^2) dx
= √(1 + 1/x^4) dx
Substituting y = 1/x, we get
A = 2π∫[1,4] (1/x) √(1 + 1/x^4) dx
This integral is not easy to solve analytically, but we can approximate it numerically using a computer or calculator. Using a calculator, we get:
A ≈ 9.73
Therefore, the exact area of the surface obtained by rotating y = 1/x about the x-axis is approximately 9.73 square units.
b) To find the area of the surface obtained by rotating y = (x^2 + 1)^(1/2) about the x-axis, we use the same formula:
A = 2π∫[a,b] y ds
where ds = √(1 + (dy/dx)^2) dx is the infinitesimal arc length, and a and b are the limits of integration.
Here, y = (x^2 + 1)^(1/2), so dy/dx = x/(x^2 + 1)^(1/2). Therefore,
ds = √(1 + (x/(x^2 + 1)^(1/2))^2) dx
= √(1 + x^2/(x^2 + 1)) dx
= √(2x^2 + 1)/(x^2 + 1) dx
Substituting y = (x^2 + 1)^(1/2), we get
A = 2π∫[0,6] (x^2 + 1)^(1/2) √(2x^2 + 1)/(x^2 + 1) dx
This integral is also not easy to solve analytically, but we can approximate it numerically using a computer or calculator. Using a calculator, we get:
A ≈ 166.54
Therefore, the exact area of the surface obtained by rotating y = (x^2 + 1)^(1/2) about the x-axis is approximately 166.54 square units.
To learn more about arc length visit;
brainly.com/question/16403495
#SPJ11
Related Questions
The percent P Of United States Residents Who were born in another country during the period from 1910 to 2000.Can be modeled by the equation p=4.8+0.165t-160 where t is the number of since 1910.
During the period of 1910 to 2000 In approximately what years? Did foreign-born residents Account for13% of all residents?
According to the model that foreign-born residents account for 4% of all residents at any time during the period 1910 to 2000 Explain your answer.
Answers
Solving the absolute value equation, it is found that:
The percentage was of 13% during these following years: 1920 and 2019.The percentage was never of 4%.What is the absolute value equation?The absolute value equation for this problem is given as follows:
p(t) = 0.165| t - 60 | + 4.8.
In which the variables are:
t is the number of years since 1910.p(t) is percentage of foreign born residents in t years after 1910.This percentage was of 13% when:
p(t) = 13.
Hence:
p(t) = 0.165| t - 60 | + 4.8.
13 = 0.165| t - 60 | + 4.8.
0.165| t - 60 | = 8.2.
|t - 60| = 8.2/0.165
|t - 60| = 49.7.
Then the solutions are as follows:
t - 60 = -49.7 -> t = 10.3 -> During the year of 1920.t - 60 = 49.7 -> t = 109.7 -> During the year of 2019.For the percentage of four percent, we have that:
4 = 0.165| t - 60 | + 4.8.
0.165| t - 60 | = -0.8.
The absolute value equation can never assume a negative value, hence the percentage will never be of 4%.
Missing InformationThe equation is:
p(t) = 0.165| t - 60 | + 4.8
More can be learned about the absolute value equation at https://brainly.com/question/17795998
#SPJ1
The product of two numbers is 36.
The sum of the numbers is 13.
What are the two numbers?
Answers
A rectangular garden has length and width as given by the expressions below.
Length: 4 - 7(3x + 4y)
Width: 3x(-2y)
Write a simplified expression for the perimeter of the rectangle.
Answers
2[4 - 21x -28y] + 2[-6xy]
8 - 42x - 56y -12xy
the perimeter of given rectangular garden is (8 - 42x - 56y -12xy).
Professor Walker notices that students in his class with larger shoe sizes tend to have higher grade point averages (GPA). Based on this observation, what is the best description of the relationship between shoe size and grade point average
Answers
There appears to be a positive correlation between shoe size and grade point average in Professor Walker's class, suggesting that students with larger shoe sizes tend to have higher GPAs.
The observation made by Professor Walker suggests a relationship between shoe size and grade point average. A positive correlation means that as one variable (shoe size) increases, the other variable (GPA) also tends to increase. In this case, the observation indicates that students with larger shoe sizes tend to have higher GPAs. However, it is important to note that correlation does not imply causation.
It is possible that the observed relationship is coincidental or influenced by other factors. Factors such as age, gender, or socioeconomic status could be associated with both shoe size and academic performance, leading to a spurious correlation. To establish a definitive causal relationship, further research and analysis would be required to account for confounding variables and determine the underlying mechanisms, if any, between shoe size and GPA.
To learn more about correlation.
Click here:brainly.com/question/30116167?
#SPJ11
13. 30 to 90
A. 3%; increase
B. 67% increase
C. 200%: increase
D. 300%; increase
Answer:
67% increase
Answers
Answer:
C
Step-by-step explanation:
The formula for percent increase is [ new-old/old * 100% ]
90-30/30 * 100%
60/30 * 100%
2 * 100%
200%
Best of Luck!
How can 10% of 82 be used to determine 60% of 82? Enter your answers in the boxes to correctly complete the statement. 10% of 82 is , so 60% of 82 is .
Answers
Answer:
To find one tenth of a number, simply divide it by 10 . The easiest way to do that is to : cross off a zero at the end if there is one. move the decimal one place to the left.
Sarah scored 82 points on her first test of the year. On the second test,
she scored 77 points. What is the percent of change?
Answers
Sarah scored 82 points on her first test of the year. On the second test,
82%
On the Second Test, She scored 77 points
77%
82
-77
=5
The Percent of Change is 5%
or -5% Because The second test Is lower than her first test
Answer:
5%
Step-by-step explanation:
82% - 77%=5%
82%
77%
so there fore the answer is 5%
How many ways can five of the letters of the word ALGORITHM be selected and written
in a row if the first two letters must be TH?
Answers
There are 210 ways to write 5 of the letters of the word ALGORITHM in a row if the 2 two letters must be TH
Given data
the word ALGORITHM
the first two letters must be TH
How to find number of ways 5 letters of the word AGORITHM will be selected and written in a row if the first two letters must be THALGORITHM has 9 letters and order of writing the letters is important, so we use Permutation to solve this
The 2 letters TH are fixed and this reduces the number of letters in the word from 9 to 7. Also, the the first two being fixed reduces the letters arrangement to be 3.
This means that we have the arrangement as follows:
7P3
= 7! / ( 7 - 3 )!
= 7! / 4!
= 7 * 6 * 5 * 4! / 4!
= 7 * 6 * 5
= 210 ways
Hence there are 210 ways to write 5 of the letters of the word ALGORITHM in a row if the 2 two letters must be TH
Read more on permutations here: https://brainly.com/question/1216161
#SPJ1
Volume of Prisms. I need help with finding the volume please
Answers
The volume of the prism is 259 units³.
How to find the volume of a prism?The volume of the prism can be calculated as follows:
volume of a prism = base area × height
The prism is a triangular base prism. The area of the base of the prism is 37 units² and the height of the prism is 7 units.
Therefore,
Base area of the prism = 37 units²
height = 7 units
Hence,
volume of a prism = 37 × 7
volume of a prism = 259 units³
learn more on volume here: https://brainly.com/question/1578538
#SPJ1
For the spring picnic, Mr. Jones served 85 hot dogs to 50 people. Mr. Jones expects 68 people at the fall picnic. Which proportion could be used to find how many hot dogs Mr. Jones needs for the fall picnic?
Answers
If Mr. Jones served 85 hot dogs to 50 people the proportion of hot dogs is:
\(\frac{85}{50}\text{ hot dogs/person}\)If he expects 68 people at the fall picnic, the amount of hot dogs he will need is the number of people multiplied by this proportion:
\(\text{hot dogs = 68p }\cdot\frac{85}{50}\text{ hot dogs/p}\)Answer
The proportion that could be used to find how many hot dogs Mr. Jones will need for the fall picnic is 85/50
He will need 115.6 hot dogs (since the number of hot dogs is a whole number, He will have to buy 116 hot dogs)
describe at least 1 way to solve linear equations? plss for 100, and brainiest.
Answers
Answer:
★ For example, Graphing is one of the simplest ways to solve a system of linear equations, all you have to do is graph each equation as a line and find the points where the lines two intersect.
Step-by-step explanation:
Hope you have a great day :)
WILL GIVE BRAINLISEST AND 60 POINTS !!!!
Question 5(Multiple Choice Worth 4 points)
(04.04)
Which graph represents a linear function?
Answers
Answer:
3rd one
Step-by-step explanation:
it is the only one with a function and a straight line
evaluate the integral. π/2 csc(t) cot(t) dt π/4
Answers
The value of the integral ∫(π/2 to π/4) csc(t) cot(t) dt is -√2 + 1.
To evaluate the integral ∫(π/2 to π/4) csc(t) cot(t) dt, we can use trigonometric identities and integration techniques.
First, let's rewrite the integrand using trigonometric identities:
csc(t) = 1/sin(t)cot(t) = cos(t)/sin(t)Substituting these identities, the integral becomes:
∫(π/2 to π/4) (1/sin(t)) * (cos(t)/sin(t)) dt
Now, we can simplify the expression:
∫(π/2 to π/4) (cos(t)/sin²(t)) dt
To evaluate this integral, we can use the substitution method. Let u = sin(t), then du = cos(t) dt. We need to find the new limits of integration when t = π/2 and t = π/4.
When t = π/2, u = sin(π/2) = 1.
When t = π/4, u = sin(π/4) = 1/√2.
The integral becomes:
∫(1 to 1/√2) (1/u²) du
Simplifying further, we have:
∫(1 to 1/√2) u^(-2) du
Now, we can integrate:
∫(1 to 1/√2) u^(-2) du = [-u^(-1)] evaluated from 1 to 1/√2
Evaluating the definite integral, we have:
[-u^(-1)] from 1 to 1/√2 = [-(1/√2)^(-1) - (-1)^(-1)] = [-√2 - (-1)] = -√2 + 1
Therefore, the value of the integral ∫(π/2 to π/4) csc(t) cot(t) dt is -√2 + 1.
For more questions on Integral
https://brainly.com/question/31433890
#SPJ11
what would happen if 300 people were sampled instead of 200, and the confidence level remained the same?
Answers
If 300 people were sampled instead of 200, and the confidence level remained the same, it would produce a more accurate result.
Sampling means selecting the group that you will actually collect data from in your research. For example, if you are researching the opinions of students in your university, you could survey a sample of 100 students. In statistics, sampling allows you to test a hypothesis about the characteristics of a population.
This is because a larger sample size allows for a better representation of the population, providing a more accurate result. Additionally, with a larger sample size, the confidence interval of the sample would be narrower, indicating a higher level of confidence in the accuracy of the result.
To learn more about sampling refer :
https://brainly.com/question/6979326
#SPJ11
A bakery offers a sale price of $2.60 for 6 muffins. What is the price per dozen?
Answers
Answer:
$5.20
Step-by-step explanation:
A dozen is 12, so multiply by two.
2.60 * 2 = $5.20
Answer:
5.20 for 12
Step-by-step explanation:
2.60 for 6
2.60*2=12
5.20 for 12
Xy has 10 pets: 4 dogs, 3 cats, 2 alpacas and 1 bunny. If he wants to arrange them in a row and make sure the pets are always grouped according to species, how many ways can he arrange the pets?
Answers
The number of ways pets can be arranged in group according to the species is in a row 288 ways.
Pets can be grouped by permutation concept as follows.
Total number of pets = 10
Number of dogs = 4
Number of cats = 3
Number of alpacas = 2
Number of bunny = 1
The number of ways pets can be arranged in group according to the species in a row as,
⇒ (4!)(3!)(2!)(1!)
⇒ (24)(6)(2)(1)
⇒ 288
Hence we can conclude that the number of ways pets can be arranged in group according to the species in a row is 288 ways.
Learn more about permutation here
https://brainly.com/question/22444718
#SPJ4
The shape is composed of three squares and two semicircles. Select all the expressions that correctly calculate the perimeter of the shape.
Answers
The expression that correctly calculates the perimeter of the shape is given as follows:
P = 2(6s + πr).
In which:
s is the side length of the square.r is the radius of the semicircle.How to obtain the perimeter of the square?The perimeter of a square of side length s is given as follows:
P = 4s.
Hence, for three squares, the perimeter is given as follows:
P = 3 x 4s
P = 12s.
How to obtain the perimeter of a semi-circle?The perimeter, which is the circumference of a semicircle of radius r, is given by the equation presented as follows:
C = πr.
Hence the perimeter of two semicircles is given as follows:
C = 2πr.
How to obtain the perimeter of the shape?The perimeter of the entire shape is given by the sum of the perimeter of each shape, hence:
P = 12s + 2πr.
P = 2(6s + πr).
More can be learned about the perimeter of a shape at https://brainly.com/question/24571594
#SPJ1
help please!
Enter an equivalent expression using the fewest number of terms.
Answers
81x²- 1 is the equivalent expression for (9x+1)(9x-1)
How to write an equivalent expression?Equivalent expressions are expressions that work the same way even though they look different
Given: the expression (9x+1)(9x-1)
Clear the parenthesis by using 9x and 1 to multiply (9x-1) separately:
(9x+1)(9x-1) = 9x(9x-1) + 1(9x-1)
= 81x²-9x +9x -1 (Add/subtract like terms)
= 81x²- 1
Therefore, the equivalent expression for (9x+1)(9x-1) is 81x²- 1
Learn more about equivalent expressions on:
https://brainly.com/question/15775046
#SPJ1
One in a thousand people have a particular disease, and the test for the disease is 98% correct in testing for the disease. On the other hand, the test has a 1% error rate, if the person being tested, does not have the disease. If someone tests positive, what are the odds they have the disease
Answers
The odds a person has the disease they were tested for is 98% or 98:2.
What is the difference between probability and odds?Probability describes how likely is an event to happen by using a percentage from 0 to 100%. On the other hand, the odds focus on the rate between the specific outcome: other outcomes.
What are the odds in this situation:In this case, the specific outcome is to have the disease after being tested, which occurs 98% of the time. This implies only the 2% tested do not have the disease. Now, this relationship can be expressed as:
98: 2
Learn more about odds in: https://brainly.com/question/545794
Find the missing angle measures.
What is m∠2?
a 40
b 60
c 50
d 100
Answers
if i get 85% on my final that's worth 10% of my grade and i currently have 85%- what will my grade be?
Answers
The final grade you will have using a mixture word problem is 85%.
What is a mixture word problem?The mixture word method or mixture word problem is a method in mathematics to solve the problem when creating a mixture of two or more different things. This method will solve the problem of determining the quantity of mixture result.
In the question, we will mix two grades. So,
= (current grade * current worth grade) + (final exam grade * final exam worth grade)
= (85% * (100%-10%)) + (85% * 10%)
= 76.5% + 8.5%
= 85%
Thus, your final grade doesn't change if you get 85% on a final exam.
Learn more about mixture word problem here:
brainly.com/question/13217705
#SPJ4
Find the optimum strategies for player A and player B in the game represented by the following payoff matrix. Find the value of the game. -7 1 2 0-4 What is the optimum strategy for player A? Choose the correct answer below, and fill in the answer box(es) to complete your choice. O A. Choose row 1 with probability Choose row 2 with probability OB. There is a saddle point, so player A will choose row (Type integers or simplified fractions.) What is the optimum strategy for player B? Choose the correct answer below, and fill in the answer box(es) to complete your choice. O A. Choose column 1 with probability Choose column 2 with probability OB. There is a saddle point, so player B will choose row (Type integers or simplified fractions.) What is the value of the game? The value of the game is (Type an integer or a simplified fraction)
Answers
A dominant strategy is a strategy that is always better than any other available strategy, regardless of the other player's choice.
To determine the optimum strategies for player A and player B in the game represented by the given payoff matrix, we need to analyze the matrix and identify any dominant strategies or saddle points.
Looking at the payoff matrix:
-7 1
2 0
-4
We can observe that there is no dominant strategy for either player A or player B. A dominant strategy is a strategy that is always better than any other available strategy, regardless of the other player's choice.
Next, we check for a saddle point, which occurs when the maximum value in a row is equal to the minimum value in the corresponding column. In this case, we find that there is no saddle point either.
When there are no dominant strategies or saddle points, the players may employ mixed strategies. In other words, they can choose a probability distribution over their available strategies.
Therefore, without additional information or specific criteria for optimizing the strategies, we cannot determine the optimum strategies for player A and player B. Furthermore, since there is no saddle point, the value of the game cannot be determined in this case.
To learn more about matrix click here
brainly.com/question/28180105
#SPJ11
if x doesn't equal 0 and x/2=y^2 and x/4=4y then what is x
Answers
The value of x is 128, for equations x/2=y² and x/4=4y
What is Equation?Two or more expressions with an Equal sign is called as Equation.
Given,
x/2=y²...(1) and
x/4=4y ...(2)
We need to find value of x and x not equal to zero
x=2y²
Substitute in equation 2
2y²/4=4y
y²/2=4y
y/2=4
y=8
Now substitute in equation 2
x/4=4(8)
x/4=32
x=128
Hence the value of x is 128, for equations x/2=y² and
x/4=4y
To learn more on Equation:
https://brainly.com/question/10413253
#SPJ1
which of the following is true for the t test for independent means?
Answers
The following is true for the t test for independent means:T test for independent means is a parametric inferential test used for making comparisons between two groups' means.
The t test for independent means can only be performed if there are two different groups or conditions.The t test for independent means assumes that the data are approximately normally distributed and that the variances of the two groups are equal.The t test for independent means can be used to examine whether or not there is a significant difference between the means of the two groups.The t test for independent means' null hypothesis is that there is no significant difference between the means of the two groups. If the t statistic obtained from the test is greater than the critical value, the null hypothesis will be rejected, indicating that there is a significant difference between the means of the two groups.The t test for independent means is an excellent way to compare two groups when the data are approximately normally distributed and the variances are equal.
Learn more about independent means here:
https://brainly.com/question/31522717
#SPJ11
Phoebe bought 2/3 of a yard of green ribbon and 1/3 of a yard white ribbon to use on her quilt. She decided to exchange 1/3 yard of white ribbon for 1/3 yard of yellow ribbon. How much white ribbon does phoebe have to use on her quilt?
Answers
Simplify the expression 5 + (15 + x).
A. 5x + 20
B. x + 20
C. x + 5
D. x+ 15
Answers
Answer:
the answer is B
I hope you a good day!
Answer:
B
Step-by-step explanation:
You can't do anything in the parentheses because you can't add x to 15 yet since x is unknown. Thus, you add 5 to 15 and get 20 + x or B, x + 20
PLEASE HELP ME :)
At a track meet, you time a friend running 100 m in 11.00 seconds. The officials time her as well. You find that you have an error of 3.09%. What are the two possible official times your friend ran the 100 meters?
Answers
Answer:
sorry but i need the answer too pls answer the question
USE A MODEL Without advertising, a Web site had 96 total visits. Today, the owners of the site are starting a new promotion, which is expected to double the total number of visits to their Web site every 5 days. a. Write an equation that relates the total number of visits, v, to the number of days the promotion has been running, d.
Answers
The equation that relates the total number of visits to the number of days the promotion has been running is v = 96 × \(2^{\frac{d}{5} }\) .
What is an equation?When an equal sign connects two expressions then, it is called an equation.
According to the question, the total number of visits doubled after every 5 days.
So, after 5 days visits= 96×2
After 10 days visits= 96×2×2
After 15 days visists= 96×2×2×2
Therefore, it is a proportional sequence.
Hence, the equation that relates the total number of visits, v, to the number of days the promotion has been running, d comes out to be:
v = 96 × \(2^{\tfrac{d}{5} }\)
Learn more about proportional equations here:
https://brainly.com/question/29864115
#SPJ1
Math question on how to evaluate f(5)
Answers
The value of f(5) in the function are - 9 . Option B
What is function?A function can be defined as an expression, rule or law that shows the relationship between a dependent variable or independent variable.
Given the functions;
f(x) = 3x - 1 (1)f(x) = 1 - 2x (2)To determine the value f(5), substitute the value of x as 5
x > 1 , so we use equation 2
From function (2)
f(5) = 1 - 2x
f(5) = 1 - 2(5)
f(5) = 1 - 10
f(5) = - 9
Thus, the value of f(5) in the function are - 9 . Option B
Learn more about functions here:
https://brainly.com/question/2328150
#SPJ1
At the p.e. class, pupils help each other measure heights. the average height of jeremy and justin is 147 cm. jeremy is 24 cm taller than justin. what is the height of jeremy?
Answers
The height of Jeremy is 159 cm.
What is average?In layman's terms, an average is a single number chosen to represent a set of numbers, typically the sum of the numbers divided by the number of numbers in the set (the arithmetic mean). The average of the numbers 2, 3, 4, 7, and 9 (summed to 25) is 5, for example. An average could be another statistic, such as the median or mode, depending on the context.To find the height of Jeremy:
The formula of average = sum of terms/number of termsLet, Justin, be x and Jeremy be x + 24So,
147 = x + (x+24) / 22x + 24 = 147 × 22x + 24 = 2942x = 294 - 242x = 270x = 135Then, Jeremy = 135 + 24 = 159 cm.
Therefore, the height of Jeremy is 159 cm.
Know more about average here:
https://brainly.com/question/20118982
#SPJ4