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
4. Molly has 20 tomato plants to arrange in her garden. What are all the ways Molly can arrange the tomato plants in an array?
Answers
The number of ways Molly can arrange the tomato plants in an array is 27,907,200 ways
Find the image attached. From the figure, we can see that there are 6 rows in the array.
If there are 20 tomato plants, the ways Molly can arrange the tomato plants in an array is expressed according to the permutation formula:
20P6 = 20!/(20-6)!
20P6 = 20!/14!
20P6 = 20 * 19 * 18 * 17 * 16 * 15
20P6 = 27,907,200
Hence the number of ways Molly can arrange the tomato plants in an array is 27,907,200 ways
Learn more on permutation formula here: https://brainly.com/question/12468032
Find the x- and y-intercepts of the parabola y=x2+4x−6.
Answers
Write the expression for the following situation: Roger has 5 toy cars. His sister gives
himxmore for his birthday. He wants to find his totd
Answers
This shows the total of Roger’s toy car plus what his sister gives.
write the equation of the line that has a slope of O and passes through the point (6, -8).
Answers
Answer:
y=-8
Step-by-step explanation:
y-y1=m(x-x1)
y-(-8)=0(x-6)
y+8=0
y=0-8
y=-8
On which number line -9 and its opposite shown?
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Answer:C TRUST MEEE
Step-by-step explanation:
mark as brainliest
M is the midpoint of the segment. Find the length of XM.
Answers
Answer:
Step-by-step explanation:
4x+3=3x+5
x+3=5
x=2
Subsitute: 4x+3=11
XM=11
Determine the equation of a circle with center at (8, 10) and radius 6.
1) (X + 8)2 + (y - 10)2 = 36
2) (x + 8)2 + (y + 10)2 = 36
3) (x-8)2 + (y - 10)2 = 36
4)(x - 8)2 + (y - 10)2 = 6
Answers
Answer:
Step-by-step explanation:
general equation for a circle at the origin is
x² + y² = r²
if the circle center is (x₀, y₀), the formula is
(x - x₀)² + (y - y₀)² = r²
(x - 8)² + (y - 10)² = 36
Determine if the sequence below is arithmetic or geometric and determine the common difference/ ratio in simplest form
15, 11 ,7, …
Answers
Answer:
This is the arithmetic sequence which has a common difference that equals to -4.
Step-by-step explanation:
An arithmetic sequence is a sequence with common difference. A common difference can be found by subtracting previous term with next term.
A common difference can be expressed in [tex]\displaystyle \large{d=a_{n+1}-a_n}[/tex] where d stands for common difference.
________
Moving to the question given. We have a sequence with given 15,11,7, ... first subtract 15 with 11.
11-15 = -4
7-11 = -4
Therefore, we can conclude that the common difference is -4 thus making the given sequence an arithmetic.
Let me know if you have any question so regarding the sequence!
are 1/2 and 2/4 proportional?
Answers
Answer:
yes they are proportional
Write in expanded form
1) 531.029
Answers
Answer:
expanded for everything
Step-by-step explanation:
Expanded Form of 531.029:
531.029 = (5 * 102) + (3 * 101) + (1 * 100) + (2 * 1/102) + (9 * 1/103)
531.029 = (5 * 100) + (3 * 10) + (1 * 1) + (2 * 1/100) + (9 * 1/1,000)
531.029 = 500 + 30 + 1 + 2/100 + 9/1,000
531.029 = 500 + 30 + 1 + 0.02 + 0.009
this drawing shows an isosceles triangle work out the size of angle a plz help as soon as possible thks
Answers
Answer:
70°
Step-by-step explanation:
Size of angle a = (180°-40°) ÷ 2 (angle sum of triangle; base angles of isos triangle)
= 140° ÷ 2
= 70°
Answer:
a + 40° = 180°
a t40°-40° =180°- 40°
a = 140°
Step-by-step explanation:
isosceles add up to 180°
Divide 3/8 divided by 2/3 and express the answer in the simplest terms.
Answers
If 11% of a number equals 12, find 66% of that number
Answers
Answer: 72
Step-by-step explanation:
0.11*a = 12
0.66*a = ?
Hint: We don't need to find a to answer this question! Just find the relationship with these equations.
0.66*a = 6*(0.11*a) = 6*12 = 72
Find all real values of $x$ such that $x^2-5x + 4=0$.
Answers
Answer:
x=4 && x=1
Step-by-step explanation:
x^2 - x - 4x + 4=0
x(x-1) - 4(x - 1)=0
(x-4)(x-1)=0
X=4 and x=1
All real values of $x$ such that $x²-5x + 4=0$ will be x=4 and x=1.
What is the equation?An equation is a statement that two expressions, which include variables and/or numbers, are equal. In essence, equations are questions, and efforts to systematically find solutions to these questions have been the driving forces behind the creation of mathematics.
It is given that, the quadratic equation is, x²-5x+4=0
Any equation of form ax²+bx+c=0 where x is variable and a, b, and c are any real numbers where a ≠ 0 is called a quadratic equation.
We have to find the value of x as,
x²-5x + 4=0
x² - x - 4x + 4=0
x(x-1) - 4(x - 1)=0
(x-4)(x-1)=0
X=4 and x=1
Thus, all real values of $x$ such that $x²-5x + 4=0$ will be x=4 and x=1.
Learn more about the equation here,
https://brainly.com/question/10413253
#SPJ6
PLZ HELPPPPP QUICKLYYYY!!!!!!!!!!!!!
Answers
Answer:
root 490
Step-by-step explanation:
help with this please
Answers
y=-2*0+12 —> y=0+12 —> y=12
so you would put that 12 in the first box. that gives you the points (0,12) on the graph and that’s where you would put your first dot…
you repeat this process for all the other x values
I don’t understand this please help
Answers
Answer:
i think BC = 8.5
Step-by-step explanation:
19-(2+6.5+2)
19-10.5
8.5
what two numbers multiply to -36 and adds to 1?
Answers
End of Semester test geometry semester a question 29
Triangle DEF is dilated with respect to the origin by a scale factor of to produce ΔD'E'F'. What is the length of side D'E'?
A.
B.
C.
D.
Answers
The resulting coordinate will be D'(2, 6), E'(4, -6), and F'(0, 10),
Let the coordinate of the triangle DF be given as D(1, 3), E(2, -3) and F(0, 5).
For instance, if the coordinate point (x, y) is dilated by a factor of k, the resulting coordinate will be (kx, ky)Note that each of the coordinates is multiplied by the scale factor.
For the coordinates D(1, 3), E(2, -3), and F(0, 5), assuming the coordinate point is dilated by a factor of 2, the resulting coordinate will be D'(2, 6), E'(4, -6), and F'(0, 10),
Learn more on dilation here: https://brainly.com/question/10253650
Note that the question is answered on a general term.
Answer:
√5
Step-by-step explanation:
I got it right on Edmentum
Hook me up with a 5 star and a thanks :)
A group of friends were working on a student film that had a budget of $1300. They used $546 of their budget on props. What percentage of their total budget did they spend on props?
Answers
546/1300 x 100 = 42%
Oceanside Bike Rental Shop charges 13 dollars plus 8 dollars an hour for renting a bike. Tim paid 45 dollars to rent a bike. How many hours did he pay to have the bike checked out?
Answers
Step-by-step explanation:
Shop charges =13
Many left for renting bike = 45-13 =$32
If renting for 1hr cost $8
then renting for X Hrs will cost $32
1/X =8/32 cross multiplying
8X= 32
X= 32÷8= 4hrs
During a chess match there are 15 pieces left on the board. the ratio of the white pieces to black pieces is 3 : 2 how many white pieces are on the white board?
Answers
==========================================================
Explanation:
The ratio 3:2 scales up to 3x:2x for some positive whole number x.
We have 3x white pieces, 2x black pieces, and 3x+2x = 5x pieces total. Set this equal to the 15 mentioned and solve for x.
5x = 15
5x/5 = 15/5
x = 3
Therefore, there are 3x = 3*3 = 9 white pieces and 2x = 2*3 = 6 black pieces, which gives 9+6 = 15 total. The ratio 9:6 reduces to 3:2 after dividing both parts by the GCF 3.
Simplify \frac{10a^3}{55a^2} when a=3.
Answers
Answer:
(27000)(27225)
Step-by-step explanation:
Answer:
6/11
Step-by-step explanation:
Captain Brewster captured a pack of aliens. The following data points represent the lengths of each of the aliens' big toes (in centimeters). 18, 16, 17, 17, 16, 15, 16. Using this date, create a dot plot where each dot represents a big toe.
Answers
Answer:
15 ·
16 ···
17 ··
18 ·
19
Step-by-step explanation:
There were 2 aliens that each have big toes measuring 17 cm long. So, the 17 would have 2 dots above it:
17 ··
help me please ill mark as the brainiest if correct.
Answers
Answer:
14 minutes
Step-by-step explanation:
For every two minutes the temperature increases by 15 c. since 8 equals 55, 10 is 70, 12 is 85, and 14 is 100.
Answer:
14
Step-by-step explanation:
let f(x)=3x-2, g(x)=5x+1
fogof(2)
Answers
Answer:
f(g(2)) = 29
Step-by-step explanation:
First of all we write the new function defined as the composite function of f and g: f(g(x)) = 3(5x+1) - 2 = 15x + 3 - 2 = 15x -1
Now we can simply "plug in" 2 instead of the x in the function we just obtained: f(g(2)) = 15*2 - 1 = 29
Measurements should ALWAYS include units in conversion problems?
A. True
B. False
Answers
The answer would be
A. True
The larger I grow, the less you can see!
What am I??
Answers
Answer: The darkness is the correct answer
Step-by-step explanation:
As you know that if the darkness becomes bigger you are unable to see it. Hope you like the answer do let me know in the about your thoughts.
Find the least common multiple of 8y^2 and 7m
Answers
Answer: [tex]56y^2m[/tex]
which is the same as writing 56y^2m
====================================================
Explanation:
Let's focus on the coefficients 8 and 7 for now.
To find the LCM of those values, list out the multiples. Circle the smallest number that can be found in both sets at the same time.
multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, 64, ...multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56, 63, ....We see that 56 is the LCM of 7 and 8.
Or you could use this shortcut
LCM = (x*y)/GCF
where x and y are the two numbers. The mention of "GCF" refers to the GCF of x and y. In this case, the GCF is 1 so,
LCM = (x*y)/GCF = (8*7)/1 = 8*7 = 56.
----------------------
Once we determine that, we look at the variable terms now.
The y^2 and m will be tacked onto the 56 to arrive at the final answer 56y^2m
This is because y and m are the unique variables, and we go for the highest exponent of each. It's similar to the LCM formula used earlier.
2. Find the equation of the perpendicular bisector of line PQ where the co-ordinates of P and Q are P (-2, 8) and Q (4,7)
Answers
Answer:
2y - 12x = 3
Step-by-step explanation:
gradient of line PQ= (7-8)/ 4+2
-1/6
gradient of perpendicular bisector of line PQ=
-1 ÷ -1/6
-1× -6
6
coordinates of the midpoint of line PQ
(-2+4)/2, (8+7)/2
(1, 15/2)
the perpendicular bisector passes through the midpoint of line PQ
Equation of the perpendicular bisector
y - 15/2 = 6(x - 1)
multiply through by 2
2y - 15 = 12(x - 1)
2y - 15 = 12x - 12
2y - 12x = 3