Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = x2, x = y2; about y = 1
Answers
D? y
= VE
Now we find the points of intersection first: 27?
-
VE
2 = 0, 1 so when x is O y is also 0
and when x is 1, y is also 1
Hence the points of intersection are (0,0) and (1,1)
Given it is rotated about y
=1
So using disk method, the volume is given by:
The volume of the solid obtained by rotating the region bounded by the given curves about [tex]y = 1[/tex] is [tex]\frac{11\pi}{30} [/tex] cubic units.
Procedure - Determination of the volume of a solid of revolutionLet [tex]f(x) = 1[/tex], [tex]g(x) = x^{1/2}[/tex] and [tex]h(x) = x^{2}[/tex], both [tex]g(x) [/tex] and [tex]h(x)[/tex] have the following intersection points: [tex](x_{1}, y_{1}) = (0, 0)[/tex] and [tex](x_{2}, y_{2}) = (1,1)[/tex]. This means that the interval of x-coordinates for the solid of revolution is [tex]x \in [0, 1][/tex].
The integral formula for the volume of the solid of revolution ([tex]V[/tex]), in cubic units, is:
[tex]V = \pi \int\limits^{1}_{0} {[h(x)-f(x)]^{2}} \, dx - \pi \int\limits^{1}_{0} {[g(x)-f(x)]^{2}} \, dx [/tex] (1)
After substituting on each function and simplifying terms we get the following integral:
[tex]V = \pi \int\limits^{1}_{0} {(x^{2}-1)^{2}} \, dx - \pi \int\limits^{1}_{0} {(x^{1/2}-1)^{2}} \, dx [/tex]
[tex]V = \pi\left[\int\limits^{1}_{0} {x^{4}} \, dx - 2\int\limits^{1}_{0} {x^{2}} \, dx -\int\limits^{1}_{0} {x} \, dx + 2\int\limits^{1}_{0} {x^{1/2}} \, dx \right][/tex]
And the volume of the solid of revolution is:
[tex]V = \pi \cdot \left(\frac{1}{5}-\frac{2}{3} -\frac{1}{2}+\frac{4}{3} \right)[/tex]
[tex]V = \frac{11\pi}{30} [/tex]
The volume of the solid obtained by rotating the region bounded by the given curves about [tex]y = 1[/tex] is [tex]\frac{11\pi}{30} [/tex] cubic units. [tex]\blacksquare[/tex]
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Related Questions
As a bonus question on a test, Mr. Smith wrote 8z + (5y6w). He told his students the values of each variable: w=6, y=2, and z = 7. What is the simplification of Mr. Smith's expression?
Answers
Answer:
416
Step-by-step explanation:
Plug in the variables
8(7) + (5*2 * 6*6)
56 + (10 * 36)
56 + 360
416
Answers:
A. 113 cm^3
B. 141 cm^3
C. 565 cm^3
D. 126 cm^3
Answers
Answer:
b
Step-by-step explanation:
What equation represents this sentence? 5 less than a number is 12. n−5=12 5=12−n 5=n−12 5−n=12
Answers
Answer:
n-5=12
This is the answer
The algebraic equation n - 5 = 12 represents the sentence " 5 less than a number is 12" and the solution of the equation is the value of variable n is 17.
What is Algebra?Algebra is the estimation of mathematical representations, while logic is the manipulation of those symbols.
The given statement is "5 less than a number is 12"
According to the given question, translate the phrase into an algebraic form :
⇒ n - 5 = 12
Thus, the algebraic equation n - 5 = 12 represents the sentence " 5 less than a number is 12"
Now the above equation solves for variable n,
⇒ n - 5 = 12
Add 5 to both sides of the above equation,
⇒ n = 12 + 5
⇒ n = 17
Therefore, the algebraic equation n - 5 = 12 represents the sentence " 5 less than a number is 12" and the solution of the equation is the value of variable n is 17.
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A washer and dryer cost 809. The washer is 59 dollars more than the dryer how much does the dryer cost
Answers
Answer:
The dryer cost $350.
Step-by-step explanation:
Answer:
$375
Step-by-step explanation:
809 - 59 = 750
750 / 2 = 375
(5 1/4−2 1/2)+(6 2/3−3 1/2) plsss help (real answers)
Answers
Answer:
5 11/12. hope this helps
Step-by-step explanation:
Answer:
6
Step-by-step explanation: 5 1/4 - 2/12 = 3 6 2/3 - 3 1/2=3 3+3=6
what is the greatest common factor of
8x - 12
Answers
Answer:
4
Step-by-step explanation:
8x - 12
4(2x - 3)
I need help I don’t understand this
Answers
Answer:
i need help I 6understand
Solve the equation for the indicated variable.
8x + y =5, for x
X= [?]
(Simplify your answer.Use integers or fractions for any numbers in the expression.)
Answers
Answer:
x = −1/8y + 5/8
Step-by-step explanation:
Step 1: Add -y to both sides.
8x = −y + 5
Step 2: Divide both sides by 8.
8x/8 = −y+5/8
x= −1/8y + 5/8
199 hares of two different colors, gray or white, gathered on a meadow. Every hare found the ratio of the number of hares of the other color to the number of hares of its color (including itself). What is the greatest value for the sum of all of these ratios?
Answers
Answer:
grey
Step-by-step explanation:
the greatest sum is 25
Need help with geometry problem
Answers
Answer:
y = [tex]6\sqrt{5}[/tex]
Step-by-step explanation:
x^2 + z^2 = 29^2
x^2 = y^2 + 9^2
y^2 + 20^2 = z^2
29^2 - z^2 = y^2 + 81 (equating for x^2 from eqn 1 and 2)
substitute 3rd eqn in 4th
29^2 - (y^2 + 20^2) = y^2 + 81
841 - y^2 - 400 = y^2 + 81
441 = 2y^2 + 81
360 = 2y^2
y^2 = 180
y = [tex]\sqrt{180}[/tex]
= [tex]\sqrt{36*5} \\[/tex]
y = [tex]6\sqrt{5}[/tex]
I need help with number 5 please
Answers
Answer:
-4[tex]\frac{1}{2}[/tex]
Step-by-step explanation:
I hope this helps!
50 points! don't answer if you don't know! Thank You
#1 Jeanne's bank account earns interest annually. The equation shows her starting balance of $200 and her balance at the end of five years, $267.65. At what rate r did Jeanne earn interest?
r= %
(Type an integer or decimal rounded to the nearest tenth as needed.)
#2 The volume of a soap bubble is 62.3mm^3 Find the radius and diameter of the soap bubble. Use 3.14 for pie.
The radius of the bubble is about mm.
(Round to the nearest tenth as needed.)
Answers
Answer:
1. 6%
2. what is the radius of 2
Step-by-step explanation:
Answer:
#1 = Rate = 0.025
#2 = m^2
Step-by-step explanation:
#1
1. starting balance = 200
balance at the end of 4 years = 220.82
2. 200.82 = 200(1+r)^4
220.82
________ = (1 + r)^4
200
1 + r = (1.1041)^1/4
1 + r = 1.025 - 1
r = 0.025
Rate = 0.025
#2
62.3m⋅m^3=62.3m^4
3.14=62.3m^4
m⋅m=m^2
Hope this helps, have a nice day/night! :D
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Pete worked 44 hours and 30 minutes on his new job. He makes $9.86 per hour. For each hour of overtime, he earns time and a half. How much will he earn for one week?
Answers
:)
Answer:
$460.955 ≈ $460.96
Step-by-step explanation:
A work week is 8 hours × 5 days = 40 hours per week. Pete worked 44 hours and 30 minutes which is 44.5 hours. This tells you that Pete worked 40 hours at regular paid and 4.5 hours of overtime at a rate is his amount per hour plus half of his amount per hour). So Pete earns $14.79 per hour for overtime rate ($9.86 + $4.93 = $14.79) .
Earning for the week: 40 × $9.86 + 4.5 × $14.79
= $394.40 + $66.555
= $460.955 ≈ $460.96
Four times the product of a number x and a number y is equal to 20.
Answers
Step-by-step explanation:
4x×y=20
.....
..................................
The expression for four times the product of a number x and a number y is equal to 20 is 4xy = 20.
Given,
Four times the product of a number x and a number y is equal to 20.
We need to write the expression.
What is the meaning of the mathematical terms?- 2 times M = 2 x M
- Product of M and N = M x N
- 3 more than M = M + 3
- 3 less than M = M - 3
Find the expression.
We have,
Four times the product of a number x and a number y is equal to 20.
Here,
The product of a number x and a number y can be written as:
= (x) x y
Four times the product of a number x and a number y can be written as:
= 4 x [ (x) x y ]
- is equal to 20 can be written as:
= 20
Now,
We get,
4xy = 20
Thus the expression for Four times the product of a number x and a number y is equal to 20 is 4xy = 20.
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What is the answer to :
2x (4+3) -5 x [4-(9-7)]
??
Answers
Answer:
8×+6×-5×{2}
8×+6×-10×
14×-10×
4×
calculation: 2x(4+3)-5x[4-(9-7)]
2x × 7-5x [4-(9-7)]
14x - 5x [4-(9-7)]
14x - 5x [2]
14x -5x × 2
14x - 10x
4x
5. The recursive algorithm given below can be used to compute gcd(a, b) where a and b are non-negative integer, not both zero.
procedure gcd(a, b)
if a b then gcd(a, b) := = gcd (b, a)
else if a = 0 then gcd (a, b) = b
else if a = 1 then gcd (a, b) :
:=1
else if a and b are even then gcd(a, b) 2gcd(a/2, b/2) := else if a is odd and b is even then gcd(a, b) := gcd(a, b/2)
else gcd(a, b) := gcd(a, b - a)
Use this algorithm to compute
(a) gcd(124, 244)
(b) gcd (4424, 2111).u
Answers
Implementating the given algorithm in python 3, the greatest common divisors of (124 and 244) and (4424 and 2111) are 4 and 1 respectively.
The program implementation is given below and the output of the sample run is attached.
def gcd(a, b):
#initialize a function named gcd which takes in two parameters
if a>b:
#checks if a is greater than b
return gcd (b, a)
#if true interchange the Parameters and Recall the function
elif a == 0:
return b
elif a == 1:
return 1
elif((a%2 == 0)and(b%2==0)):
#even numbers leave no remainder when divided by 2, checks if a and b are even
return 2 * gcd(a/2, b/2)
elif((a%2 !=0) and (b%2==0)):
#checks if a is odd and B is even
return gcd(a, b/2)
else :
return gcd(a, b-a)
#since it's a recursive function, it recalls the function with new parameters until a certain condition is satisfied
print(gcd(124, 244))
print()
#leaves a space after the first output
print(gcd(4424, 2111))
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The largest frog in the world is the goliath, It can grow to be 12 inches long. The smallest frog in the world is about 2.5% as log as the goliath. About how long is the smallest frog in the world
Answers
Answer:
0.3 inches
Step-by-step explanation:
2.5% = 0.025
The smallest frog is 12 * 0.025 inches, which is 0.3 inches.
I need help with this problem! please hurry have 3 more after so let me know if you want to do them to and ill give 200 points.
Answers
Last year 85 students in 7th grade This school year only 68 What is the percent decrease?
Answers
Answer: 11.11%
17:153*100 =
(17*100):153 =
1700:153 = 11.11
If an amount of $25,000 is deposited into a saving account at an interest rate 4%, compounded semi-annually, how long will it take to accumulate on this account $28,500? The function describing your balance is M(n) =25,000(1+0.04/2)^2n, where n is time in years. 20 points and WILL GIVE BRAINLIEST!
Answers
It will take 3.3 years to accumulate $28,500 on this account
According to the compound amount formula expressed as:
[tex]A=P(1+\frac{r}{n} )^{nt}[/tex]
P is the principal = $25,000
A is the amount = $28,500
r is the rate in decimal = 4% = 0.04
n = 2 (semi annualy)
Substitute the given parameters into the formula as shown:
[tex]28,500=25,000(1+\frac{0.04}{2} )^{2t}\\1.14=(1+0.02)^{2t}\\1.14 = (1.02)^{2t}\\ln1.14 = 2tln1.02\\2t=\frac{ln1.14}{ln1.02}\\2t= \frac{0.13103}{0.01980}\\2t= 6.6177\\t = 3.30yrs[/tex]
Hence it will take 3.3 years to accumulate $28,500 on this account
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10
Find the area of the shape.
Either enter on exact answer in terms of it or use 3.14 for it and enter your answer as a decimal.
units
Show Calculator
Answers
Answer:
(4-x)(x-4)=16-x2 help me please
Which of the functions below have period pie check all that apply
Answers
Help help help help help help help help
Answers
Answer:
7
Step-by-step explanation:
15-8 is 7
16-8 is 7
17-10 is 7
18-11 is 7
this fits the eqation.
hope this helps you
WHAT DOES EXPONENT C MEAN LETS SAY A NUMBER IS IN PARANTHESIS AND WITHIN THAT PARENTHESIS THERES A C AS AN EXPONENT WHAT DOES THAT MEAN
Answers
Answer:
It means that the number is raised to the power of c. For example, if it is (2^3), the answer would be 8.
Step-by-step explanation:
Find the measure of
Answers
Answer:
∠ A = 43°
Step-by-step explanation:
Since the triangles are similar then corresponding angles are congruent, so
∠ B = ∠ T = 55°
The sum of the 3 angles in a triangle = 180°
sum the 3 angles in Δ ABC and equate to 180
2x - 9 + 3x + 4 + 55 = 180 , that is
5x + 50 = 180 ( subtract 50 from both sides )
5x = 130 ( divide both sides by 5 )
x = 26
Then
∠ A = 2x - 9 = 2(26) - 9 = 52 - 9 = 43°
A man is 32 years older than his son ten year ago he was three times as old as his son was then find the present age of each?
Answers
x-10=3(y-10)=3y-30
X+20=3y
3x-96=3y
Thus, X+20=3x-96, and 2x=116, X=58
Now, 58–32=y, thus, y=26
What is the first step needed to solve 3 over 4 multiplied by x minus 3 equals negative 18 ? (1 point) a Add 3 to both sides b Subtract 18 from both sides c Multiply both sides by 3 d Divide both sides by 4
Answers
Answer:
a
Step-by-step explanation:
3/4 x - 3 = -18
You should try to get all of one term on one side of the equation, and have the constant on the other side so the next step would be easy.
Here are the steps after you add 3:
3/4 x = -15
x = -5/4 (divide by 3/4 on both sides)
Answer is in the screenshot:
Find the missing number in the following series: __ 85 81 83 79 81 77
Answers
Answer:
83
Step-by-step explanation:
The pattern is -4 then +2
If
x
=
2
and
y
=
5
, evaluate the following expression:
2
(
3
y
−
4
x
)
Answers
Are the ratios 2:1 and 11:9 equivalent?
Answers
Answer: No
Step-by-step explanation:
2 is double 1.
Because 11 is not double 9, the ratios are not equivalent.
Hope it helps :)