The volume of a solid revolution generated by rotating the curve y = f(x) and x = f(y) between x = a and x = b , y = a and y = b through 360 degrees about the x-axis and y-axis is given
[tex]V_{x} =\int\limits^b_a {\pi y^{2} } \, dx[/tex] and [tex]V_{y} =\int\limits^b_a {\pi x^{2} } \, dy[/tex]
The diagram shows the line y = 1, the line y = 4 and part of the curve [tex]y=3x^2[/tex]. The shaded region is rotated through 360 degrees about the y-axis. Find the exact value of the volume of revolution obtained. Leave your answer in pi.
Answers
Answer:
[tex]\dfrac{5}{2}\pi[/tex]
Step-by-step explanation:
Rotation about the y-axis
[tex]\textsf{Volume}=\displaystyle \int^b_a \pi x^2\:\text{d}y[/tex]
where:
b = upper limita = lower limitx is a function of yGiven function of y: [tex]y = 3x^2[/tex]
Rewrite the given function as a function of y:
[tex]\implies x^2=\dfrac{1}{3}y[/tex]
Substitute the values into the formula:
[tex]\implies \displaystyle \int^4_1 \dfrac{1}{3}\pi y\:\:\text{d}y[/tex]
[tex]\boxed{\begin{minipage}{5 cm}\underline{Terms multiplied by constants}\\\\$\displaystyle \int ay^n\:\text{d}y=a \int y^n \:\text{d}y$\end{minipage}}[/tex]
[tex]\boxed{\begin{minipage}{4 cm}\underline{Integrating $y^n$}\\\\$\displaystyle \int y^n\:\text{d}y=\dfrac{y^{n+1}}{n+1}+\text{C}$\end{minipage}}[/tex]
Take out the constant and integrate:
[tex]\begin{aligned}\implies \dfrac{1}{3}\pi\displaystyle \int^4_1 y\:\:\text{d}y & = \dfrac{1}{3}\pi \left[\dfrac{1}{2}y^2\right]^4_1\\\\& =\dfrac{1}{3}\pi \left[\dfrac{1}{2}(4)^2-\dfrac{1}{2}(1)^2\right]\\\\&=\dfrac{1}{3}\pi\left[8-\dfrac{1}{2}\right]\\\\&=\dfrac{5}{2}\pi \end{aligned}[/tex]
Therefore, the exact value of the volume of revolution is:
[tex]\dfrac{5}{2}\pi[/tex]
Learn more about integration here:
https://brainly.com/question/27988986
https://brainly.com/question/27805589
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Step-by-step explanation:
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Step-by-step explanation:
Given data
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f(x)=3x+10
Hence the cost of 8 movies is
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Step-by-step explanation:
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Answer:
Step-by-step explanation:
Dividing by a fraction is the same as multiplying by the upside-down fraction.
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Answers
Answer:
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Step-by-step explanation:
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Step-by-step explanation:
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Answer:
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Step-by-step explanation:
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Me pueden ayudar porfis no entiendo
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Answer:
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Answer: A and D
Step-by-step explanation: I got it right
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Answer:
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Step-by-step explanation:
hope it helps!
....
Answer:
501 × 502 = 251,502
...
Can someone help me please
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Answer:
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Step-by-step explanation:
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Step-by-step explanation:
math question file down below
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Answers
Note: Steps of student are not given. So, the proper steps of the solution are provided below.
Given:
The inequality is
[tex]-5v>55[/tex]
To find:
Where the student made any errors?
Solution:
We have,
[tex]-5v>55[/tex]
On dividing both sides -5 change the sign of equality because -5 is a negative number.
[tex]\dfrac{-5v}{-5}<\dfrac{55}{-5}[/tex]
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Answers
Answer:
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Step-by-step explanation:
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Step-by-step explanation:
Answer:
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Step-by-step explanation:
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Step-by-step explanation:
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WILL GIVE BRAINLIST!!!!
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Step-by-step explanation:
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asdfruhyhkijgfdsafuhyfjtfrdef
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stop random spam(thanks for points xD)
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Answers
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Step-by-step explanation:
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To obtain the value of one unknown quantity linear equation in one variable can be formed. The cost of one mug is given as $21.
What is a Linear equation?A linear equation is a equation that has degree as one.
To find the solution of n unknown quantities n number of equations with n number of variables are required.
Given that,
The cost of 3 calculators = The cost of 7 mugs.
The cost of each calculator is $12 more than each mug.
Suppose, cost of each mug be x.
And, cost of each calculator is x + 12.
As per the question the following equation can be made,
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To know more about linear equation click on,
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Step-by-step explanation:
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Step-by-step explanation:
i dont know this one but an easy was to get all of your answers is to search up something like exponent calculator and i use sights like mathpapa and caculator soup soo good luck