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
Pls help with this:/
Answers
9514 1404 393
Answer:
a. y = -1/3x +7/3
b. f(6) = 1/3
Step-by-step explanation:
The two given points are (1, 2) and (4, 1). The slope of the line between them is given by the slope formula:
m = (y2 -y1)/(x2 -x1)
m = (1 -2)/(4 -1) = -1/3
The point-slope form of the equation of the line is ...
y -k = m(x -h) . . . . . . line with slope m through point (h, k)
We can add k to both sides of this equation and substitute for our slope (-1/3) and the first point (1, 2). This gives ...
y = m(x -h) +k
y = -1/3(x -1) +2
y = -1/3x +7/3 . . . . eliminate parentheses
__
b. The value of y when x=6 is ...
y = -1/3(6) +7/3 = (-6 +7)/3
y = 1/3
That is, for f(x) = -1/3x +7/3, ...
f(6) = 1/3
write the following numbers as the product of a natural number with a power of 10
a) 2450
b) 35000
c)4000400
d) 10101010
Answers
[tex]\\ \sf\longmapsto 2450=2\times 35\times 35\times 1^{10}[/tex]
#2
[tex]\\ \sf\longmapsto 35000=35\times 10(10)(10)\times 1^{10}[/tex]
#3
[tex]\\ \sf\longmapsto 4000400=2\times 2\times 1001\times 100\times 1^{10}[/tex]
#4
[tex]\\ \sf\longmapsto 10101010=1010101\times 10\times 1^{10}[/tex]
Step-by-step explanation:
The numbers are given in the form of N*10ⁿ, where N is a natural number
a) 2450 = 245*10 b) 35000 = 35*10³ c)4000400 = 40004*10² d) 10101010 = 1010101*10Louis is comparing the populations of several nearby cities. The population of Liberty Crossing can be written as 49 people, and the population of Harrisburg can be written as 47 people. The population of Glenview is 43 times the population of Harrisburg. Drag the cities to put them in order from least population (top) to greatest population (bottom).
Answers
Step-by-step explanation:
Louis is comparing the populations of several nearby cities. The population of Liberty Crossing can be written as 49 people, and the population of Harrisburg can be written as 47 people. The population of Glenview is 43 times the population of Harrisburg. Drag the cities to put them in order from least population (top) to greatest population (bottom).
The order of the cities from least population to greatest population:
Harrisburg
Glenview
Liberty Crossing
Given the information provided:
Harrisburg: This city has a population of 47 people. It's the smallest population among the three cities.
Glenview: The population of Glenview is 43 times the population of Harrisburg.
Since Harrisburg has 47 people, the population of Glenview would be 43 x 47 = 2021 people.
So, Glenview has a larger population than Harrisburg but is still smaller than Liberty Crossing.
Liberty Crossing: The population of Liberty Crossing is given as 49 people, which is the largest population among the three cities.
So, the correct order from least population to greatest population is:
Harrisburg
Glenview
Liberty Crossing
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Complete the sentences to make true statements
6 is 100 times ____.
0.06 is 10 times ____.
60 is 1/100 of ____.
Answers
Answer:
6 is 100 times 0.06
0.06 is 10 times 0.006
60 is 1/100 of 6,000
Step-by-step explanation:
Since 6/100 is equal to 0.06, we can say that 6 is the result of multiplying 0.06 by one hundred (0.06 x 100 = 6).
On the other hand, since 0.06 / 10 is equal to 0.006, it is correct to say that 0.006 x 10 is equal to 0.06 (0.006 x 10 = 0.06).
Finally, 60 is one hundredth of 6000, since multiplying 60 x 100 the result is 6,000 (6,000 / 100 = 60).
hope this really helped!!
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The complete sentences are:
6 is 100 times 0.06 .
0.06 is 10 times 0.006.
60 is 1/100 of 6,000.
"6 is 100 times greater than 0.06":
This means that if you multiply 0.06 by 100, you get 6.
So, 6 is indeed 100 times larger than 0.06.
"0.06 is 10 times greater than 0.006":
In this case, if you multiply 0.006 by 10, you get 0.06.
Thus, 0.06 is 10 times larger than 0.006.
"60 is 1/100 of 6,000":
This means that if you divide 6,000 by 100, you get 60.
So, 60 is a hundredth (1/100) of 6,000.
Hence, 6 is 100 times 0.06
0.06 is 10 times 0.006
60 is 1/100 of 6,000
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The points (-1, 4) and (2, 9) lie on a line.
What is the slope of the line?
Answers
Answer:
[tex]\displaystyle m = \frac{5}{3}[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightAlgebra I
Coordinate Planes
Coordinates (x, y)Slope Formula: [tex]\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}[/tex]Step-by-step explanation:
Step 1: Define
Identify
Point (-1, 4)
Point (2, 9)
Step 2: Find slope m
Simply plug in the 2 coordinates into the slope formula to find slope m.
Substitute in points [Slope Formula]: [tex]\displaystyle m = \frac{9 - 4}{2 + 1}[/tex][Order of Operations] Simplify: [tex]\displaystyle m = \frac{5}{3}[/tex]-7(8) What is it please help me I need help help help help
Answers
find the degree of the polynomial 5ab^3-7a^5c^2-5
Answers
Answer:
7
Step-by-step explanation:
as -7a^5c^2 term has two cofficient a and b and each have power.by adding power of one term we get 7
Help!!!!! I will give out brainiest to best answer!!!
Answers
Answer:
see below
Step-by-step explanation:
a = 4
b = -2
c = -8
d = -11
Answer:
a=4, b=-2, c=-8, and d=-11
Step-by-step explanation:
its pretty easy
A shirt that normally cost $30 is on sale for $21.75 what percent of the regular price is the sale price
Answers
Answer: 72.5%
Step-by-step explanation:
I used trial and error to figure this out.
I first found a quarter of 30 which is 7.5, so the percentage is near 75% since when 7.5 is times by 3 it becomes 22.5 which is close. Then I found what one is by dividing 30 by 100 and guessed the percentages getting closer to the answer basing on whether my last answer was to low or high and found 72.5% of 30 is 21.75 so the answer is 72.5%.
8 m good wire how long is the wire in millimeters
Answers
Answer:
Converting m to mm
=> 1 m = 1000 mm
=> 8 m = 8000 mm
Therefore,
The wire will be 8000 mm long.
Hope it helps you...
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a
Use a trigonometric equation to
determine the leg of this triangle.
10 m
[?] m
30°
Answers
Answer:
use pythogoras therom to solve it and its say to find base its formula is h square - psquare
For the two numbers listed, find two factors of the first number such that their product is the first number and their sum is the second number -12,-1
Answers
-4 x 3 = -12
-4 + 3 = -1
6+7+8+...+(n+5) find the sum.
Answers
Answer:
n + 26
Step-by-step explanation:
first you have to simplify
6 + 7 +8 + n + 5
then combine the like terms
6 + 7 + 8 + n + 5
(n) + ( 6 + 7 + 8 + 5)
= n + 26
hope this helps
what is the measure of angle x
Answers
Answer:
180
Step-by-step explanation:
Find the value of X in triangles by subtracting known angle measurements from 180 degrees. Since the value of all angles within a triangle must equal 180 degrees, if you know at least two angles, you can subtract them from 180 to find the missing third angle.
5 lb 6 oz = __________ oz
Explain yourself
Answers
Answer:
86 oz
Step-by-step explanation:
1 lb = 16 oz
5 lb * 16 oz = 80 oz
80 oz + 6 oz = 86 oz
Start with x and cube it
Answers
Answer:
x to the power of 3
Step-by-step explanation:
What is an equivalent expression for 1.5(37* 4) * 0.25(Qr+8)
Answers
Answer:
-4 I believe is the answer
Step-by-step explanation:
Hope this helps:)
If Esteban continues at the same rate, how many miles will he have traveled in 6 hours?
A. 36
B. 300
C. 360
D. 420
Answers
Answer360
Step-by-step explanation:
If u=log(x3+y3+z3-3xyz)show that d2u/dx2+d2u/dy2+d2u/dz2=-3/(X+y+z)^2
Answers
What is the expression 3/8d+3/4 factored using the GCF
Answers
Answer:
[tex]\frac{3}{8}[/tex] ( d + 2)
Step-by-step explanation:
[tex]\frac{3}{8}[/tex] d + [tex]\frac{3}{4}[/tex]
= [tex]\frac{3}{8}[/tex] d + [tex]\frac{6}{8}[/tex] ← factor out [tex]\frac{3}{8}[/tex] from each term
= [tex]\frac{3}{8}[/tex] (d + 2) ← in factored form
PLZ HELP
What question should you ask yourself first when factoring polynomials?
A: Can I divide all terms by a GCF?
B: What is the exponent of the first term?
C: What is the degree of the polynomial?
D: Are there four terms?
Answers
Answer:
The correct answer is A: Can I divide all terms by a GCF?
explanation:
I took the quiz
1/8 of a pumpkin pie has 379 calories. How many calories
are in the whole pie?
Answers
Answer:
379*8
Step-by-step explanation:
1. 300*8=2400
2.70*8 = 560
3. 9*8=72
3032
A random number generator generates numbers from 1 to 10. What is the probability that the number generator generates a 3 ?
Answers
Answer:
1/10
Step-by-step explanation:
Sarah had 1 3/4 bags of trail mix. She age 1/2 of a bag. She gave 1/3 of a bag to her friend. She dropped 1/6 of a bag and she was sad. How much trail mix did she had left?
Answers
Answer:
3/4 bags of trail mix
Step-by-step explanation:
She ate 1/2 of a bag, she gave 1/3 of a bag, and she dropped 1/6 of a bag. In total she "lost" 1/2+1/3+1/6=1 bag of trail mix. So she has 1 3/4-1=3/4 bags left
the independent and dependent variables.
1) You are taking a road trip in a car and want to know how far you've traveled (read the odometer) at various times
during your trip.
A) does not describe a function
B) Independent variable, distance traveled; dependent variable, time
C) Independent variable, time; dependent variable, distance traveled
actont Writon statement that describes how one
Answers
Answer:
C) Independent variable, time; dependent variable, distance traveled
Explanation:
An independent variable (also known as the manipulated variable) is the thing you change on purpose. A dependent variable (also known as responding variable) is what you measure on different kinds of trials. You are measuring how far you've gone. You will measure this at various times (you change this).
Correct answer is C : Independent variable, time; dependent variable, distance traveled actont Writon statement that describes how one.
What is Independent variable?The independent variable exists as the variable being regulated in the problem, and the dependent variable exists as the variable that varies because of this control.
It exists a variable that stands alone and isn't altered by the other variables you exist trying to measure.
An independent variable (also known as the manipulated variable) is the thing you change on purpose. A dependent variable (also known as responding variable) is what you measure on different kinds of trials. You are measuring how far you've gone. You will measure this at various times (you change this)To learn more about Independent variable refer to:
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42. A football team is at its own 25-yard line. In the next three plays, it loses an average of 4 Yards per play. Where is the team after the three plays?
Answers
4 yards per play * 3 plays = 12 total yards lost
25 - 12 = 13
Find the 20 th term of the given sequence 2, 5, 8, 11,..
Answers
Answer:
14,17, 20
Step-by-step explanation:
Kindly see the picture I have solved it as required
What is the inequality graphed below?
Answers
Answer:
y>=5/3x+5
Step-by-step explanation:
line is y=5/3x+5
shading is above the line
line is shaded
y>=5/3x+5
what is the diameter of a circle that has the circumference of 400
Answers
Answer:
400/π
Step-by-step explanation:
Well, we know that the formula for circumference is πd. (d is the diameter) Then, we set up the equation 'πd = 400'. Now, simplifying for d, we get d = 200/π.
Answer:
400/π
Step-by-step explanation:
Graph the number on the number line.
Answers
Answer:
Step-by-step explanation:
can u take good picture