Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
[tex]y = {x}^{2} [/tex]
[tex]y = 4x[/tex]
; about the y-axis

Answers

Answer 1

well, first off, when it comes to volumes by rotation, we'd want to graph them, Check the picture below. Our rotation over the axis will give us a "washer", so we'll be using the washer method.

now, our axis of rotation is the y-axis or namely x = 0, x = 0 is a vertical line, meaning we have to put the functions in y-terms, that is in f(y) form

[tex]y = x^2\implies \pm\sqrt{y}=x\implies \boxed{\pm\sqrt{y}=f(y)} \\\\\\ y = 4x\implies \cfrac{y}{4}=x\implies \boxed{\cfrac{y}{4}=g(y)}[/tex]

if we look at the picture, the parabola is the farthest from the axis of rotation and the line is the closest, or namely R² and r² respectively.

the way I get the area for R² and r², is by using the same I do with "area under the curve", so if I say call the axis of rotation h(y), the way I get the area is

R => f(y) - h(y)

r => g(y) - h(y)

so let's proceed.

[tex]\textit{area under the curve}\\ \begin{array}{llll} \sqrt{y}- 0\implies &\sqrt{y}\\ \frac{y}{4}-0\implies &\frac{y}{4} \end{array}\qquad \qquad \begin{array}{llll} \stackrel{R^2}{(\sqrt{y})^2}-\stackrel{r^2}{\left( \frac{y}{4} \right)^2}\\[-0.5em] \hrulefill\\ y\qquad -\qquad \frac{y^2}{16} \end{array}[/tex]

now, we want to get the area enclosed by both, and thus we'd need their points of intersection, setting both to [tex]\sqrt{y}=\cfrac{y}{4}[/tex] which in short gives us the bounds of 0 and 16.

[tex]\pi \displaystyle\int_{0}^{16}\left( y - \cfrac{y^2}{16} \right)dy\implies \pi \int_{0}^{16}y\cdot dy-\pi \int_{0}^{16}\cfrac{y^2}{16}\cdot dy\\\\\\\pi \cdot \left. \cfrac{y^2}{2} \right]_{0}^{16}-\pi \cdot \left. \cfrac{y^3}{48} \right]_{0}^{16}\implies 128\pi -\cfrac{256\pi }{3}\implies \boxed{\cfrac{128\pi }{3}}[/tex]

Find The Volume V Of The Solid Obtained By Rotating The Region Bounded By The Given Curves About The

Related Questions

Someone help, thanks

Answers

Answer:

i speak very little english sorry

Step-by-step explanation:

You buy 3.18 pounds of apples, 1.35 pounds of oranges, and 2.25 pounds of pears. What is your total bill?

Answers

Answer:

6.78

Step-by-step explanation:

Answer:

6.78 pounds

Step-by-step explanation:

Add: 3.18+1.35+2.25=6.78

(If you wanted the cost for the whole thing then you need to include the price for each of the pounds)

x/2 - 3 = 5 Solve the equation algebraically show work to support the answer

Answers

Answer:

x = 16

Step-by-step explanation:

Answer:

x = 16

Step-by-step explanation:

x/2 - 3 = 5

x/2 = 5+3

x/2 = 8

x = 8×2

x = 16

6. The California Tiger Salamander is an endangered species, which decreases at the rate of 4.6% per year in a habitat that currently has 60 of them. Write an exponential function and find how many California Tiger Salamanders will be left after 4 years.

7. Factor and solve the following equation 2x2 + x - 21 = 0.

8. Alvin throws the football to a receiver who jumps up to catch the ball. The height of the ball over time can be represented by the quadratic equation -4.9t2 + 7.5t + 1.8 = 2.1 . This equation is based on the acceleration of gravity -4.9 m/s2, the velocity of his pass is 7.5 m/s and releases the football at a height of 1.8 meters, and the height where the receiver catches the ball of 2.1 meters. Put the equation in standard form and then solve by using the quadratic equation.

9. Examine the graph of f(x) and the table that contains values of g(x). Which function has a greater rate of change over the interval [0,2]? Explain your answer.

Answers

[tex]\rule{2000pt}{1000000pt}[/tex]

Write -37.75 as a mixed number.

Answers

Answer:

[tex]-37\frac{3}{4}[/tex]

Step-by-step explanation:

0.75 is equivalent to three quarters, or (3/4).

3/4 would be the fraction. -37 would be the whole number.

[tex]-37.75 = \boxed{-37\frac{3}{4}}[/tex]

Hope this helps.

Answer:

- 37 3/4

Step-by-step explanation:

MAX POINTS!!! Where did I go wrong? Please explain where I went wrong and redo it.

Answers

Answer:

Ms. I am sure that you are already correct

(4 − 3)(−3 + 6) − (2 − )
simplify
show all work

Answers

Answer:

(4 − 3)(−3 + 6) − (2 )

1(3)-2

3-2

Hope This Helps!!!

The verticles of a triangle are the points R(3,c),Q(9,2) and R(3c,11) where c is constant. Given that angle PQR is 90

Answers

Answer:

Step-by-step explanation:

Find c if ∠PQR = 90°?

I will ASSUME you mean point P is at (3. c)

slope of PQ is (2 - c) / (9 - 3) = (2 - c) / 6

slope of QR is (11 - 2) / (3c - 9) = 9 / (3c - 9)

perpendicular lines have negative reciprocal slopes.

(2 - c) / 6 = -1(3c - 9)/9

9(2 - c) = -6(3c - 9)

18 - 9c = -18c + 54

9c = 36

c = 4

Set up an algebraic equation then solve. 1. When 3 is subtracted from the sum of a number and 10 the result is 2. The sum of 3 times a number and 12 is equal to 3. Find the number. 3. Three times the sum of a number and 6 is equal to 5 times the number. Find the number. 4. The width of a rectangle is 4 inches less than its length. If the perimeter measures 72 inches, find the dimensions of the rectangle. 5. The perimeter of a square is 48 inches. Find the length of each 2. Find the number. side.

Answers

Answer:

( x+10)-3= 2

x+10= 5

x= -5

3x+12= 3

3x= -9

x= -3

3( x+6 )= 5x

3x+ 18 = 5x

2x= 18

x =9

w= l-4 y= x-4

perimeter= 2l+ 2w

2(2x) + 2( x-4)

4x+2x-8 = 72

6x= 72+8

6x= 80

x= 13.3 width = 9.3 length = 13.3

4x=48

x = 12

each 2= 24

1. Bianca can walk 5/6 of a mile in 1/3 of an hour. What is her walking speed, in miles per hour?
2. How long, in minutes, does it take Bianca to walk 1 mile?

Answers

Answer:

1. 2.5 miles/hour

2. 24 minutes

Step-by-step explanation:

1. 5/6 of mile in 1/3 of an hour. In 1 hour, Bianca can walk 5/6 x 3 = 15/6 miles or 2.5 miles/hour

2. 5/6 of mile in 1/3 of an hour, hour in 1 mile = (1 x 1/3)/(5/6) = 2/5 = 0.4 hour = 24 minutes

Image attached giving 25 points please help

Answers

Answer:

Rational

Step-by-step explanation:

It isnt whole because its a decimal,

It isnt an Integer because its a decimal

It isnt natural because its a decimal

Meaning it can only be rational

A firm has the following fixed proportion production function q = min(k, 2l). a. Find the firms long-run total, average, and marginal cost functions. b. Suppose that k is fixed at 20 in the short run. Calculate the firms short-run total, average, and marginal cost functions. c. Suppose v = 1 and w = 3. Calculate this firms long-run and short-run average and marginal cost curves. Q4. A firm’s production function is given by q = 2z1 + 3z2. Given the input prices, w1 for z1 and w2 for z2. a. Find the input demands z1 and z2 minimizing the cost of producing the level of output q. b. Find the cost function for producing the level q.

Answers

Answer:

I think its right

Step-by-step explanation:

a) In the long run, we have, 5k=10 and k=2l Thus, C=2l+3l= 5l=0.5q Thus, AC=5l/10l =0.5 MC=0.5 b) In the short run, k=10 Q=min(50,10l) If l<5, q=10l. C=10+3l= 10+0.3q Thus, AC=10/q + 0.3 If l>5, q=50. C= 10+3l AC= (10+3l)/50 If q>50, then MC...

Write a system of equations to describe the situation below, solve using any method, and fill
in the blanks

Mrs. McMillan, the PE, teacher, is pairing off students to race against each other. Heather can
run 4 meters per second, and Erin can run 5 meters per second, Mrs. McMillan decides to give Heather a head start of 23 meters since she runs more slowly. Once the students start
running, Erin will quickly catch up to Heather. How far will Erin have to run? How long will
that take?

Please help.

Answers

Answer:

23 seconds later Erin will catch up

Erin will run 115 meters

Step-by-step explanation:

23+4x=5x

x=23

23 seconds later Erin will catch up

Erin will run 115 meters

Which of the following is the graph of y = |x - 3|

Answers

Answer:

y = | x - 3 | is graphed on the attachment below ↓

Step-by-step explanation:

If you ever need an equation graphed, here's a useful site called Desmos that can do that really well and for free.

2 + 2 = 3
prove me wrong

Answers

Answer:

2+2=4................

Answer:

you are wrong because 2+2=4

solve for x. 2/x-3=x/12-4x

Answers

Answer:

x= 47 18+2 201 i ≈0.382978723+0.603295612i x= 47 −2 201 i+18 ≈0.382978723−0.603295612i Hopefully this makes just the same amount of sense as this question

PLS HELP PLEASSSSS IM SERIOUSLY CRYING RN

Answers

i really don’t know 555

[tex]x = 23[/tex]

Step-by-step explanation:

First, we need to write the equations in their standard form, i.e., all the variables and their coefficients are to be placed on the left hand side and the plain numbers on the right hand side. The first equation is already in its standard form:

[tex]2x + 5y = 35\;\;\;\;\;(1)[/tex]

but we need to rewrite the 2nd one. So the standard form of the 2nd equation can be written as

[tex]x + 15y = -10\;\;\;\;\;(2)[/tex]

Now that the equations are in their standard forms, the next step that we need to do is to eliminate one of the variables, namely the y. To do that, we need to multiply either one of the equations so that when we add them together, the y-variable gets eliminated. Note if we multiply Eqn(1) by -3, we'll get -15y on one equation and a +15y on the other equation so that when they are added together, they cancel out. So if we do that, we get

[tex]-6x - 15y = -105\;\;\;\;\;(3)[/tex]

Adding Eqn(3) to Eqn(2), we get

[tex]-5x = -115[/tex]

[tex]x = \dfrac{-115}{-5} = 23[/tex]

Tìm nghiệm tổng quát của các phương trình
y"+2y'-3y=0

Answers

Answer:

y € R i guess

Step-by-step explanation:

I hope it correct

Sandra signed up for the gym during the spring special.
Her credit card was charged $220.87. Is this the correct amount? Explain.
A sign that says workout jazz spring special. Sign up now and get 12 months for just 16 dollars and 99 cents per month asterisk. The asterisk is given as paid in advance.

No; the gym should have only charged Sandra $16.99.

No; the gym charged Sandra $16.99 less than it should have.

No; the gym charged Sandra $16.99 more than it should have.

Yes; the gym charged Sandra the correct amount.

Answers

Answer:

yes the gym charged Sandra the correct price

PLLLZZZ NO ONE WILL HELP ME!!!
Given this equation in slope-intercept form…
y = 2x + 5
Identify or solve for the following:
Slope
y-intercept
x-intercept
Independent variable
Dependent variable
Domain
Range
We would like to represent a line passing through the point (5,0) with a slope of 2. Please write the equation for this line in slope-intercept form and in point-slope form.

What is the rate of change of the function above?

Answers

Answer:

y = m x + b standard form of straight line

y = 2 x + 5

The slope is m and = 2

the y intercept is 5 because when x = o y = 5

if y = 0 then x = -2/5 and would be the x intercept

x is the independent variable and y is the dependent variable

I am not familiar with your definitions for domain and range but

both x and y can vary from -∝ to = ∝

(-7,-4) (-11,-2) slope

Answers

Answer:

1/3

Step-by-step explanation:

-2-4 = -6

-7-11 = -18

Answer them all please

Answers

Answer:

see the attachment

i hope it helps u ✌️✌️✌️☠️

a) The highest common factor and the least common multiple are 12 and 48, respectively.

b) The highest common factor and the least common multiple are 12 and 72, respectively.

c) The highest common factor and the least common multiple are 8 and 160, respectively.

d) The highest common factor and the least common multiple are 12 and 48, respectively.

e) The highest common factor and the least common multiple are 14 and 840, respectively.

Procedure - Determination of the Highest Common Factor and the Least Common Multiple of two integers

In this question we must determine the highest common factor and the least common multiple of two integers.

The highest common factor consists in the maximum product of common prime numbers, of the two numbers that divides it, and the least common multiple consists in the product of prime numbers, common and not, of the two numbers, that creates a product that it is greater or equal to the greater integer.

Now, we proceed to determine the numbers for each case:

a) 12 and 48