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
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]
Related Questions
help help help meeee
Answers
Answer:
122
Step-by-step explanation:
since line m and line n are parallel the sum of the angles is equal to 180 degrees
58 + x = 180 subtract 58 from both sides
x = 122
PLS HELP PLEASSSSS IM SERIOUSLY CRYING RN
Answers
[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]
or
[tex]x = \dfrac{-115}{-5} = 23[/tex]
A rectangular athletic field is twice as long as it is wide. If the perimeter of the athletic field is 90 yards, what are its dimensions?
Answers
Answer:
L=30 W=15
Step-by-step explanation:
90 divided by 6 is 15. 15 is the Width. Multiply 15 by 2 and you get 30 which is the Length.
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
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...
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
Someone help, thanks
Answers
Answer:
i speak very little english sorry
Step-by-step explanation:
What is the standard deviation of 82,93,89 and 88?
Answers
Answer:
4.546, rounded
Step-by-step explanation:
pls mark brainliest!
(-7,-4) (-11,-2) slope
Answers
Answer:
1/3
Step-by-step explanation:
-2-4 = -6
-7-11 = -18
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
1
Hope This Helps!!!
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 = ∝
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
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 integersIn 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 48First, we factorize each integer:
[tex]12 = 2^{2}\times 3[/tex], [tex]48 = 2^{4}\times 3[/tex]
Then the highest common factor and the least common multiple are, respectively:
[tex]LCM = 2^{4}\times 3 = 48[/tex], [tex]HCF = 2^{2}\times 3 = 12[/tex]
The highest common factor and the least common multiple are 12 and 48, respectively. [tex]\blacksquare[/tex]
b) 24 and 36First, we factorize each integer:
[tex]24 = 2^{3}\times 3[/tex], [tex]36 = 2^{2}\times 3^{2}[/tex]
Then, the highest common factor and the least common multiple are, respectively:
[tex]LCM = 2^{3}\times 3^{2} = 72[/tex], [tex]HCF = 2^{2}\times 3 = 12[/tex]
The highest common factor and the least common multiple are 12 and 72, respectively. [tex]\blacksquare[/tex]
c) 32 and 40First, we factorize each integer:
[tex]32 = 2^{5}[/tex], [tex]40 = 2^{3}\times 5[/tex]
Then, the highest common factor and the least common multiple are, respectively:
[tex]LCM = 2^{5}\times 5 = 160[/tex], [tex]HCF = 2^{3} = 8[/tex]
The highest common factor and the least common multiple are 8 and 160, respectively. [tex]\blacksquare[/tex]
d) 24, 48 and 60First, we factorize each integer:
[tex]24 = 2^{3}\times 3[/tex], [tex]48 = 2^{4}\times 3[/tex], [tex]60 = 2^{2}\times 3 \times 5[/tex]
Then, the highest common factor and the least common multiple are, respectively:
[tex]LCM = 2^{4}\times 3 \times 5 = 48[/tex], [tex]HCF = 2^{2}\times 3 = 12[/tex]
The highest common factor and the least common multiple are 12 and 48, respectively. [tex]\blacksquare[/tex]
e) 42, 56 and 70First, we factorize each integer:
[tex]42 = 2\times 3 \times 7[/tex], [tex]56 = 2^{3}\times 7[/tex], [tex]70 = 2\times 5\times 7[/tex]
Then, the highest common factor and the least common multiple are, respectively:
[tex]LCM = 2^{3} \times 3 \times 5 \times 7 = 840[/tex], [tex]HCF = 2\times 7 = 14[/tex]
The highest common factor and the least common multiple are 14 and 840, respectively. [tex]\blacksquare[/tex]
To learn more on prime numbers, we kindly invite to check this verified question: https://brainly.com/question/4184435
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
Which of the following is a correct interpretation of the expression 5−8
Answers
Answer:
start at 5 on the number line and move 8 to the left.
Step-by-step explanation:
Explanation:
5 - 8 = -3
Therefore the final answer should be placed at negative three on the number line.
When you move to the left eight times away from the number five, the answer would be negative three.
Dog slept 3 hours . If the dog snored 1/3 of an hour . How many times did he snore
Answers
Answer:
it depends how much the god snores within a certain time period
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)
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:
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.
Interest = $45.58, time = 1 year, rate = 2.25%. Find the principal. Round to the nearest hundredth place
Answers
Answer:
$2,025.78
Step-by-step explanation:
I = Prt ⇒ P = [tex]\frac{I}{rt}[/tex]
2.25% = 0.0225
P = 45.58 ÷ ( 0.0225 × 1 ) ≈ $2,025.78
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
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]
Acellus. please help me
Answers
by-step explaAnswer:
by-step epla
Step-nation:
a
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
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
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
some help, I need some answers
Answers
Answer:
Well your Hypotenuse is 10 and your other angle is 8 so it is most likely 6 because you see a change in 2 each time and it is about 2 units smaller because all right triangles follow a pattern
Step-by-step explanation:
if it was too go even higher it would go to 12,14,16 etcetera
2 + 2 = 3
prove me wrong
Answers
Answer:
2+2=4................
Answer:
you are wrong because 2+2=4