Suppose f(x,y,z)=z and W is the bottom half of a sphere of radius 2 . Enter rho as rho, ϕ as phi, and θ as theta. (a) As an iterated integral, ∫∫∫WfdV=∫AB∫CD∫EF drhodϕdθ with limits of integration A = B = C = D = E = F = (b) Evaluate the integral.
Answers
a) The limits of integration for the triple integral are \(\int _0^{2\pi}\) \(\int _0^{2\pi}\) \(\int _0^{5}\) f(ρ,φ,θ) ρ²sinφ dρdφdθ
b) The value of the integral is 10π.
The limits of integration for the triple integral will depend on the volume of integration. In this case, the volume is the bottom half of a sphere of radius 5, which means that ρ varies from 0 to 5, φ varies from 0 to π/2, and θ varies from 0 to 2π. Hence, the limits of integration for the triple integral are:
\(\int _0^{2\pi}\) \(\int _0^{2\pi}\) \(\int _0^{5}\) f(ρ,φ,θ) ρ²sinφ dρdφdθ
To evaluate this integral, we need to set up a triple integral that represents the volume of the region W and the function f(x,y,z) over that region. The integral notation is represented as:
∫∫∫ f(x,y,z) dV
where dV represents an infinitesimal volume element and the limits of integration are determined by the region W. Since W is the bottom half of a sphere of radius 5, we can use spherical coordinates to represent the limits of integration.
In spherical coordinates, the volume element dV is represented as:
dV = ρ²sin(φ)dρdθdφ
where ρ represents the radial distance, φ represents the polar angle (measured from the positive z-axis), and θ represents the azimuthal angle (measured from the positive x-axis).
To integrate over the region W, we need to set the limits of integration accordingly. Since we are only looking at the bottom half of a sphere, the limits for ρ, φ, and θ are as follows:
0 ≤ ρ ≤ 5
0 ≤ φ ≤ π/2
0 ≤ θ ≤ 2π
Plugging in the limits of integration and the volume element into the integral notation, we get:
∫∫∫ f(x,y,z) dV = \(\int _0^{2\pi}\) \(\int _0^{2\pi}\) \(\int _0^{5}\) 1 / √(ρ²) ρ²sin(φ) dρdφdθ
Simplifying this expression, we get:
\(\int _0^{2\pi}\) \(\int _0^{2\pi}\) \(\int _0^{5}\) sin(φ) dρdφdθ
Evaluating the innermost integral with respect to ρ, we get:
\(\int _0^{2\pi}\) \(\int _0^{2\pi}\) 5sin(φ) dφdθ
Evaluating the middle integral with respect to φ, we get:
\(\int _0^{2\pi}\) [-5cos(φ)]dθ
Simplifying this expression, we get:
\(\int _0^{2\pi}\) 5 dθ = 10π
To know more about integral here
https://brainly.com/question/18125359
#SPJ4
Related Questions
Question 19
A quarter is dropped from the roof of a building with a height of 80 m. The following equation can be used to determine the velocity, v, in
meters per second, of an object when it hits the ground.
V= √2x9.8 xh, where h is the height of the object from the ground
What is the velocity of the quarter as it hits the ground? Round your answer to the nearest tenth.
meters/sec
Answers
The velocity of the quarter when it hits the ground is 39.6 meters per second.
A quarter is dropped from the roof of a building with a height of 80 meters. The following equation can be used to determine the velocity, v, in meters per second, of an object.
V= √(2×9.8×h)
Here, the variable "h" denotes the distance travelled by the quarter during the free-fall. We need to find out the velocity of the quarter when it hits the ground. The quarter will cover a distance equal to the height of the building when it hits the ground.
V = √(2×9.8×h)
V = √(2×9.8×80)
V = √1,568
V = 39.6
Hence, the velocity of the quarter when it hits the ground is 39.6 meters per second.
To learn more about velocity, visit :
https://brainly.com/question/14236800
#SPJ1
Given are the following data for year 1: Profit after taxes = $5 million; Depreciation = $2 million; Investment in fixed assets = $4 million; Investment net working capital = $1 million. Calculate the free cash flow (FCF) for year 1:
Group of answer choices
$7 million.
$3 million.
$11 million.
$2 million.
Answers
The free cash flow (FCF) for year 1 can be calculated by subtracting the investment in fixed assets and the investment in net working capital from the profit after taxes and adding back the depreciation. In this case, the free cash flow for year 1 is $2 million
Free cash flow (FCF) is a measure of the cash generated by a company after accounting for its expenses and investments in fixed assets and working capital. It represents the amount of cash available to the company for distribution to its shareholders, reinvestment in the business, or debt reduction.
In this case, the given data states that the profit after taxes is $5 million, the depreciation is $2 million, the investment in fixed assets is $4 million, and the investment in net working capital is $1 million.
The free cash flow (FCF) for year 1 can be calculated as follows:
FCF = Profit after taxes + Depreciation - Investment in fixed assets - Investment in net working capital
FCF = $5 million + $2 million - $4 million - $1 million
FCF = $2 million
Therefore, the free cash flow for year 1 is $2 million. This means that after accounting for investments and expenses, the company has $2 million of cash available for other purposes such as expansion, dividends, or debt repayment.
Learn more about free cash flow here:
brainly.com/question/28591750
#SPJ11
please help I need it please
Answers
Answer:
42
Step-by-step explanation:
3^2 = 9
2*9 = 18
4*6 = 24
18 + 24 = 42
Explanation: (plug-in the values) 2(3)^2 + 4(6) ; 6^2 + 24 ; 36 + 24 = 60
(Hope this helps and please mark me as brainliest!)
Can a cube of a number be a negative number? Explain?
Answers
Answer:
yes
Step-by-step explanation:
If the number is negative (ex. -2), then yes
(-2 * -2 * -2 = -8)
Highest common factor of 30 and 75
Answers
3 times 5=15
HCF15
Which expression represents a negative value?
A) -13 + 31
B -8.4
C –40 : (-5)
D-2 - (-7)
-
Answers
\(-40:(-5) \\ ⇒ \frac{ - 40}{ 5} \\ ⇒ - 8 \)
- BRAINLIEST answerer
question content area bottom part 1 a. which one of the following statements is correct concerning the concept of materiality? (a) Materiality is determined by reference to guidelines established by the AICPA
(b) Materiality depends only on the dollar amount of an item relative to other items in the financial statements
(c) Materiality depends on the nature of an item rather than the dollar amount
(d) Materiality is a matter of professional judgment.
Answers
The statement that is correct concerning the concept of materiality is , Materiality is a matter of professional judgment , the correct option is (d) .
The Materiality is defined as a concept used in auditing and accounting to determine the significance of an item or amount in financial statements.
It is a matter of professional judgment based on the auditor's assessment of the size, nature, and circumstances of the item or amount in relation to the financial statements as a whole.
Materiality is not determined by specific guidelines established by the AICPA or any other accounting or auditing standards-setting organization.
Therefore , The correct statement is represented by Option(d) .
Learn more about Professional Judgment here
https://brainly.com/question/28663304
#SPJ4
The given question is incomplete , the complete question is
Which one of the following statements is correct concerning the concept of materiality?
(a) Materiality is determined by reference to guidelines established by the AICPA
(b) Materiality depends only on the dollar amount of an item relative to other items in the financial statements
(c) Materiality depends on the nature of an item rather than the dollar amount
(d) Materiality is a matter of professional judgment.
The radius of a circle is 8 miles. What is the length of a 90 arc?
Answers
Answer:
Step-by-step explanation:
Diego is attending UTEP and has completed some courses worth 3 credits and some courses worth 4 credits. He has earned a total of 59 credits after completing 18 courses.
How many courses worth 4 credits has Diego completed?
Answers
Answer:
5
Step-by-step explanation:
5 courses worth 4 credits have Diego completed. Therefore, option B is the correct answer.
We need to find how many courses worth 4 credits has Diego completed.
What are systems of equations?In mathematics, a system of equations, also known as a set of simultaneous or an equation system, is a finite set of equations for which we sought common solutions. A system of equations can be classified in a similar manner as single equations.
Now, let the courses worth 3 credits be x and the courses worth 4 credits be y.
Now, in total Diego completed 18 courses
That is, x+y=18 ------(I)
Diego has earned a total of 59 credits
So, 3x+4y=59 ------(II)
Multiply the equation (I) by 3, that is 3x+3y=54 ------(III)
Now, subtract the equation (III) from (II) and simplify.
That is, 3x+4y-(3x+3y)=59-54
⇒y=5
Substitute y=5 in the equation (I) and simplify
That is, x=13
Therefore, 5 courses worth 4 credits have Diego completed.
To learn more about the system of equations visit:
https://brainly.com/question/21620502.
#SPJ6
Given the cost function C(x) = 0.75 + 63, 180 and the revenue
function R(x) = 1.92x, find the break-even point.
Answers
The break-even point is 54,000 units given the cost function and revenue function.
The break-even point is the moment at which a product's sales revenue equals its production costs. There is currently neither a gain nor a loss.
This problem's cost function is C(x) = 0.75x + 63,180, where x is the total number of created units. R(x) = 1.92x, where x is the quantity of units sold, is the revenue function.
We must set the revenue equal to the expense in order to determine the break-even point:
R(x) = C(x) (x)
1.92x = 0.75x + 63,180
1.17x = 63,180
x = 54,000
Thus, 54,000 units are needed to break even.
This means that the money earned will be equivalent to the cost of creating the product if the company produces and sells 54,000 units of it. The corporation will make money on any units sold over this threshold while losing money on any units sold below it.
Learn more about function here:
https://brainly.com/question/12431044
#SPJ1
Given: ∆MNP, PM = 8 m∠P = 90°, m∠N = 58° Find: Perimeter of ∆MNP
(Not 22.4 or 22.43)
Please answer ASAP, brainly awarded.
Answers
Answer:
Step-by-step explanation:
Triangle MNP is a right triangle with the following values:
m∠P = 90°m∠N = 58°PM = 8Interior angles of a triangle sum to 180°. Therefore:
m∠M + m∠N + m∠P = 180°
m∠M + 58° + 90° = 180°
m∠M + 148° = 180°
m∠M = 32°
To find the measures of sides MN and NP, use the Law of Sines:
\(\boxed{\begin{minipage}{7.6 cm}\underline{Law of Sines} \\\\$\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$\\\\\\where:\\ \phantom{ww}$\bullet$ $A, B$ and $C$ are the angles. \\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides opposite the angles.\\\end{minipage}}\)
Substitute the values into the formula:
\(\dfrac{MN}{\sin P}=\dfrac{NP}{\sin M}=\dfrac{PM}{\sin N}\)
\(\dfrac{MN}{\sin 90^{\circ}}=\dfrac{NP}{\sin 32^{\circ}}=\dfrac{8}{\sin 58^{\circ}}\)
Therefore:
\(MN=\dfrac{8\sin 90^{\circ}}{\sin 58^{\circ}}=9.43342722...\)
\(NP=\dfrac{8\sin 32^{\circ}}{\sin 58^{\circ}}=4.99895481...\)
To find the perimeter of triangle MNP, sum the lengths of the sides.
\(\begin{aligned}\textsf{Perimeter}&=MN+NP+PM\\&=9.43342722...+4.99895481...+8\\&=22.4323820...\\&=22.43\; \sf units\; (2\;d.p.)\end{aligned}\)
Easy 7th grade work sheet ! Please help , I’ll give brainliest (all answers)
Answers
2. -11
3. -6
4. 9
5. 8
6. 5
7. Simply place dots on the numbers stated, which are 11, -5, and -8.
8. The absolute value of a non-zero number is positive because absolute value is a magnitude, and is either positive or zero. The absolute value of any non-zero number can be thought of as its distance from zero, which will always be positive.
Please mark as branliest if i helped!!:)
find the area of the rectilinear
Answers
Answer:
52cm^2
Step-by-step explanation:
I did the splitting in my head and that's pretty hard to explain like this so, just BTW if your confused on where I got my lengths from.
(7*6) + (5*2)
(42) + (10)
52
I separated them into an almost square and a rectangle, then I found the area of each rectangle and then added them together.
Hopefully this was useful and you didn't have any issues with this!
the body mass of a man is xkg.thebody mass of his two children are five-sixth and four_fifths of their father5 x over 6 + 4 x over 5 5 x over 6 + 4 x over 5
Answers
56/120
Step-by-step explanation:
The body masses of the two children in terms of their father's body mass, x, are:
First child's body mass = 5x/6 kg
Second child's body mass = 4x/5 kg
To express the body mass of the man's two children in terms of their father's body mass, we can use the given ratios.
Let the body mass of the man be x kg.
The first child's body mass is five-sixths of their father's body mass:
Body mass of the first child = (5/6) * x
= 5x/6 kg.
The second child's body mass is four-fifths of their father's body mass:
Body mass of the second child = (4/5) * x
= 4x/5 kg.
Therefore, the body masses of the two children in terms of their father's body mass, x, are:
First child's body mass = 5x/6 kg
Second child's body mass = 4x/5 kg
Learn more about the ratio here:
brainly.com/question/1504221
#SPJ4
Olanda, Linda and Roberta each bought a bag of chips at a concession stand, and Marsala bought a cookie. Problem: audio What decimal is equivalent to the fraction of the girls who bought chips? Enter your answer in the box.
Answers
Answer:
0.75
Step-by-step explanation:
Given
\(Chips=\{Olanda, Linda, Roberta\}\)
\(Cookies=\{Marsala\}\\\)
Required
Equivalent of those that bought chips
From the question, 3 out of the 4 girls bought chips.
So, the fraction is:
\(Chips = \frac{3}{4}\)
Express as decimal
\(Chips = 0.75\)
Hence: the decimal equivalent is 0.75
for the quadratic function f(x)=x^2 6x 9 determine where f(x)0 and where f(x)0. select the correct choice below and fill in the answer box(es) within your choice.
Answers
Since the parabola opens upwards and the vertex is the minimum point, f(x) is greater than 0 (positive) for all x values except at x = 3, where it is equal to 0.
To determine where the quadratic function f(x) = x^2 - 6x + 9 is equal to 0, we need to find the x-values that satisfy the equation f(x) = 0.
We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula.
Using factoring:
x^2 - 6x + 9 = 0
(x - 3)(x - 3) = 0
(x - 3)^2 = 0
By factoring, we see that the equation is a perfect square trinomial, and the factor (x - 3) is repeated. Therefore, the only solution is x = 3.
So, f(x) = 0 at x = 3.
To determine where f(x) is greater than 0 or less than 0, we can analyze the shape of the quadratic curve. Since the coefficient of x^2 is positive, the parabola opens upwards, and the vertex represents the minimum point.
The vertex of the quadratic function f(x) = x^2 - 6x + 9 can be found using the formula x = -b/2a. In this case, a = 1 and b = -6.
x = -(-6) / 2(1)
x = 6/2
x = 3
The vertex is located at (3, f(3)).
Since the parabola opens upwards and the vertex is the minimum point, f(x) is greater than 0 (positive) for all x values except at x = 3, where it is equal to 0.
Therefore:
- f(x) > 0 for x ≠ 3
- f(x) = 0 at x = 3
Please note that I have assumed that the given quadratic function is f(x) = x^2 - 6x + 9, as the question does not specify the signs of the quadratic terms. If the signs are different, the results may vary.
To know more about quadratic function related question visit:
https://brainly.com/question/29775037
#SPJ11
.Find the largest subset S on which the given vector - valued function is continuous?
F = ( t/9+t^4, 7t^2-7t+1, ln (t+3))
.
Answers
The largest subset S on which the vector-valued function F is continuous is the interval (-3, ∞).
To find the largest subset S on which the given vector-valued function is continuous, we need to examine the individual component functions and identify any restrictions or conditions that could cause discontinuities.
Component 1: f1(t) = t/(9 + t^4)
This function is a rational function. It will be continuous for all real values of t except where the denominator equals zero: 9 + t^4 = 0. However, there are no real values of t that satisfy this equation, so f1(t) is continuous for all real values of t.
Component 2: f2(t) = 7t^2 - 7t + 1
This function is a polynomial function, and polynomial functions are continuous for all real values of t. Therefore, f2(t) is continuous for all real values of t.
Component 3: f3(t) = ln(t + 3)
The natural logarithm function is defined for the positive values of its argument. Therefore, t + 3 > 0, which implies t > -3. Hence, the function f3(t) is continuous for t > -3.
Taking all these considerations into account.The function is continuous for all values of t greater than -3.
Learn more about subset visit:
brainly.com/question/32263610
#SPJ11
Create ABC by drawing AC. AC represents the foreman’s line of sight to the top of the landfill. What is m
Answers
Where the above is given, the required angle m∠BAC = 45°.
In triangle ABC. AC represents the foreman’s line of sight to the top of the landfill. Landfill height is BC
What is triangle?The triangle is geometric shape which includes 3 sides and sum of interior angle should not grater than 180°
According to conditions angle b = 90°
The sum of angles of a triangle= 180°
That is a + b + c = 180
Therefore, c = a
a = (180 - b)/2
= (180 - 90) / 2
= 90 / 2
= 45°
Hence, the required angle m∠BAC = 45°
Learn more about triangles at:
brainly.com/question/273823
#SPJ1
Full Question:
Although part of your question is missing, you might be referring to this full question:
Question 1 Create Triangle ABC by drawing AC. Segment AC represents the foreman’s line of sight to the top of the landfill. What is Angle m BAC?
Please Help
Which of the following is a continuous random variable?
A. X= number of incoming flights at the local airport
B. T=winning time in the man's 100 meter dash at the 2016 Olympics
C. P=number of points scored by Stephen curry in a game
D.H=The number of hats sold on Tuesday
Answers
Answer:
option A
Step-by-step explanation:
because it is not limited to anything while option b is limited to the men's hundred meter dash and 2016, and to the Olympics
C is limited to one game and stephen curry
D is limited to tuesday
since option a isn't confined to one day, person, or specific event there can be endless numbers assigned to the variable
Write an equation to represent the following description.
Thirteen more than five times a number is – 3.
Answers
Answer:
5x+13=-3
Step-by-step explanation:
5x+13=-3
Question 10 of 28
Which of the following expressions are equivalent to
2. ? Choose all that apply.
ھر - 20
A.
B.
c.
2
(3)2 - (13 )2
1
1
(x3 - 3 ) ( + 13 )
2
1
(3 - 13) (3 - 13)
مردم زمرے
2
1
D. (x3 - 3 ) ( + 13 )
Answers
Answer: A, D
Step-by-step explanation:
A) Correct. \((x^3)^2=x^{(3)(2)}=x^6\). Similar logic for y.
B) Incorrect. The numerators are different.
C. Incorrect. The denominators are not the same.
D. Correct. The denominators are the same by difference of squares.
Bess drew a rectangular prism that is 12 units long, 8 units wide, and 13 units tall. What is the volume of the rectangular prism? A. 109 cubic units B. 156 cubic units C. 970 cubic units D. 1,248 cubic units
Answers
Answer:
D. 1,248 cubic units
Step-by-step explanation:
the volume of the rectangular prism = length*width*height = 12*8*13 = 1248 cubic units
Answer: D. 1,248 cubic units
Step-by-step explanation:
The formula for a rectangular prism volume is: V = Length x Width x Height.
In this question it already gives us all of the sides that we need! So all we need to do is just multiply all the sides together to get the volume.
12 x 8 x 13 = 1,248. Therefore D is correct!
Hope this helped you!
suppose that you and a friend are playing cards and decide to make a bet. if your friend draws two non-face cards, where a face card is a jack, a queen, or a king, in succession from a standard deck of 52 cards replacing the first card, you give him $30. otherwise, he pays you $50. if the same bet was made 20 times, how much would you expect to win or lose? round your answer to the nearest cent, if necessary.
Answers
I will win about $ 59.
This is a question of permutations and combinations.
We know that,
Probability of getting two non face cards = \(\frac{^{40}C_2}{{^{52}C_2}}\) = 10/17
Probability of not getting two non face cards = 1 - (10/17) = 7/17
Hence, we can write,
Money I will get = (7/17)*50*20 = $ 411.764
Money I'll have to pay = (10/17)*30*20 = $ 352.94
We know that,
Money left with me = Money I will get - Money I'll have to pay
Hence, we can write,
Money left with me = $ 411.764 - $ 352.94
Money left with me = 58.824 ≈ $ 59
To learn more about Probability, here:-
https://brainly.com/question/11234923
#SPJ4
which one is the right answer
Answers
Answer:
Use a website called Desmos and click on the graphing calculator
Step-by-step explanation:
Type all of those equations in the graphing calculator and the one that is the same as the graph above is the correct one!
Answer:
Since it’s negative it will have a negative slope so it would be down to a, b, c
Then look at the y intercept which is 2 and it’s positive so it would be c
Answer:C
A survey was given to a random sample of 1350 residents of a town to determine whether they support a new plan to raise taxes in order to increase education spending. Of those surveyed, 64% of the people said they were in favor of the plan. At the 95% confidence level, what is the margin of error for this survey expressed as a percentage to the nearest tenth?
Answers
The margin of error for the survey, rounded to the nearest tenth, is approximately 4.0% when expressed as a percentage.
To determine the margin of error for a survey at the 95% confidence level, we need to calculate the standard error. The margin of error represents the range within which the true population proportion is likely to fall.
The formula for calculating the standard error is:
Standard Error = sqrt((p * (1 - p)) / n)
where p is the sample proportion and n is the sample size.
In this case, the sample proportion is 64% (or 0.64) since 64% of the 1350 surveyed residents support the plan.
Plugging in the values:
Standard Error = \(\sqrt{(0.64 * (1 - 0.64)) / 1350)}\)
\(= \sqrt{(0.2304 / 1350)} \\= \sqrt{(0.0001707)}\)
≈ 0.0131
Now, to find the margin of error, we multiply the standard error by the appropriate critical value for a 95% confidence level. The critical value corresponds to the z-score, which is approximately 1.96 for a 95% confidence level.
Margin of Error = z * Standard Error
= 1.96 * 0.0131
≈ 0.0257
Finally, to express the margin of error as a percentage, we divide it by the sample proportion and multiply by 100:
Margin of Error as Percentage = (Margin of Error / Sample Proportion) * 100
= (0.0257 / 0.64) * 100
≈ 4.0%
Therefore, the margin of error for this survey, expressed as a percentage to the nearest tenth, is approximately 4.0%.
For more question on margin visit:
https://brainly.com/question/15691460
#SPJ8
34 ÷ 57 simplify the answer
Answers
Answer:
It can't be simplified anymore
Step-by-step explanation:
2 times 17 = 24
3 times 19 = 57
Answer:
brainliest plss
Step-by-step explanation:
answer is in the img belowwwww
Find the geometric mean of the numbers. If necessary, give the answer in simplest radical form.
1.5 and 20
The geometric mean is .
Answers
The following are parity questions a. Assume odd parity and add a parity bit to 11110101 b. Is the following data plus parity bit correct for even parity? Why or why not? 100011101 c. Is the following data plus parity bit correct for odd parity? Why or why not?. 011111010
Answers
a) The resulting data with the parity bit would be 111101011. b) the given data plus parity bit is not correct for even parity. c) The given data plus parity bit is not correct for odd parity.
a. To add a parity bit for odd parity to the binary number 11110101, we need to count the number of 1s in the original data. In this case, there are five 1s. Since odd parity means the total number of 1s, including the parity bit, should be odd, we need to add a parity bit with a value of 1. The resulting data with the parity bit would be 111101011.
b. The given data plus parity bit is 100011101. For even parity, the total number of 1s, including the parity bit, should be even. Let's count the number of 1s in the data: there are five 1s. Since the total number of 1s is odd, it does not satisfy the even parity requirement. Therefore, the given data plus parity bit is not correct for even parity.
c. The given data plus parity bit is 011111010. For odd parity, the total number of 1s, including the parity bit, should be odd. Let's count the number of 1s in the data: there are six 1s. Since the total number of 1s is even, it does not satisfy the odd parity requirement. Therefore, the given data plus parity bit is not correct for odd parity.
To learn more about parity here:
https://brainly.com/question/31963332
#SPJ4
Help!!!!!! pls 50 PTS
Given f(x) = 3[[x-3]] + 2, find each value
f(1.6)
f(-3.3)
Answers
A function assigns the values. The value of f(1.6) and f(-3.3) is -2.2 and -16.9, respectively.
What is a Function?A function assigns the value of each element of one set to the other specific element of another set.
Given the function f(x) = 3(x-3) + 2, now to know the value of f(1.6) and f(-3.3), subtitute the value of x as 1.6 and -3.3, respectively.
f(x) = 3(x-3) + 2
f(1.6) = 3(x-3) + 2
= 3(1.6 - 3) + 2
= 3(-1.4) + 2
= -2.2
f(-3.3) = 3(x-3) + 2
= 3(-3.3 - 3) + 2
= 3(-6.3) + 2
= -18.9 + 2
= -16.9
Hence, the value of f(1.6) and f(-3.3) is -2.2 and -16.9, respectively.
Learn more about Function here:
https://brainly.com/question/5245372
#SPJ1
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. 9. y= x
,y=0,x=4
Answers
The volume generated by rotating the region bounded by the curve y = x about the y-axis using the method of cylindrical shells is 486π cubic units.
To find the volume generated by rotating the region bounded by the curve y = x about the y-axis using the method of cylindrical shells, we can follow these steps:
First, let's sketch the region bounded by the curve y = x. This is a straight line passing through the origin with a slope of 1. It forms a right triangle in the first quadrant, with the x-axis and y-axis as its legs.
Next, we need to determine the limits of integration. Since we are rotating about the y-axis, the integration limits will correspond to the y-values of the region. In this case, the region is bounded by y = 0 (the x-axis) and y = x.
The height of each cylindrical shell will be the difference between the upper and lower curves. Therefore, the height of each shell is given by h = x.
The radius of each cylindrical shell is the distance from the y-axis to the x-value on the curve. Since we are rotating about the y-axis, the radius is given by r = y.
The differential volume element of each cylindrical shell is given by dV = 2πrh dy, where r is the radius and h is the height.
Now we can express the volume of the solid of revolution as the integral of the differential volume element over the range of y-values:
V = ∫[a, b] 2πrh dy
Here, [a, b] represents the range of y-values that define the region. In this case, a = 0 and b = 9 (as y = x, so the curve intersects y-axis at y = 9).
Substituting t
he values of r and h into the integral, we have:
V = ∫[0, 9] 2πy(y) dy
Simplifying, we get:
V = 2π ∫[0, 9] y^2 dy
Evaluating the integral, we have:
V = 2π [y^3/3] from 0 to 9
V = 2π [(9^3/3) - (0^3/3)]
V = 2π [(729/3) - 0]
V = 2π (243)
V = 486π
to learn more about integration.
https://brainly.com/question/31744185
#SPJ11