1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
djyliett [7]
2 years ago
10

A) Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given cur

ves about the x-axis.
xy = 2, x = 0, y = 2, y = 4
b) Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis.
y = x3, y = 8, x = 0; about x = 9
Mathematics
1 answer:
Leno4ka [110]2 years ago
3 0

(a) See the attached sketch. Each shell will have a radius <em>y</em> chosen from the interval [2, 4], a height of <em>x</em> = 2/<em>y</em>, and thickness ∆<em>y</em>. For infinitely many shells, we have ∆<em>y</em> converging to 0, and each super-thin shell contributes an infinitesimal volume of

2<em>π</em> (radius)² (height) = 4<em>πy</em>

Then the volume of the solid is obtained by integrating over [2, 4]:

\displaystyle 4\pi \int_2^4 y\,\mathrm dy = 2\pi y^2\bigg|_{y=2}^{y=4} = 2\pi (4^2-2^2) = \boxed{24\pi}

(b) See the other attached sketch. (The text is a bit cluttered, but hopefully you'll understand what is drawn.) Each shell has a radius 9 - <em>x</em> (this is the distance between a given <em>x</em> value in the orange shaded region to the axis of revolution) and a height of 8 - <em>x</em> ³ (and this is the distance between the line <em>y</em> = 8 and the curve <em>y</em> = <em>x</em> ³). Then each shell has a volume of

2<em>π</em> (9 - <em>x</em>)² (8 - <em>x</em> ³) = 2<em>π</em> (648 - 144<em>x</em> + 8<em>x</em> ² - 81<em>x</em> ³ + 18<em>x</em> ⁴ - <em>x</em> ⁵)

so that the overall volume of the solid would be

\displaystyle 2\pi \int_0^2 (648-144x+8x^2-81x^3+18x^4-x^5)\,\mathrm dx = \boxed{\frac{24296\pi}{15}}

I leave the details of integrating to you.

You might be interested in
A bookstore bought a case of 10 books at a wholesale price of $84. The bookstore will sell the books at a 25 percent markup, and
Nata [24]
All you have to do is multiply 84 by 25%

And that's your answer.

I hope this helped
4 0
3 years ago
Read 2 more answers
How to factor 9X^3-27
azamat
You will find teh GCF of that this will give you 9(x3-3) now you can fator it teh factors of these can be (x2+1) and (x-3)
4 0
3 years ago
Based on the figure below, what is the value of x? (5 points) A right angle is shown divided in two parts. The measure of the an
Orlov [11]

Answer:

Hello! answer: 9

Step-by-step explanation:

Assuming this is a complementary angle (meaning it will add up to 90 degrees)

5 × 9 = 45 45 + 15 = 60 60 + 30 = 90 therefore x = 9 Hope that helps!

8 0
3 years ago
Read 2 more answers
There are 40 animals at a petting zoo, all of which are goats or rabbits, and each animal is either an adult or a child. There a
Mrac [35]
This question has been posted several times, and the answer can be found @ brainly.com/question/3387068
4 0
3 years ago
Read 2 more answers
Mary eats 5 bananas ann an additional 2 bananas when she works our for 2 hours. Represent the number of bananas that she eats wh
Mila [183]

5+2+5+2=14 that is the answer


7 0
3 years ago
Other questions:
  • What is the gcf of 42 and 154
    5·1 answer
  • Maneesh and Beth are using leather to make bracelets. Maneesh has y inches of leather. Beth has 7 more inches of leather than Ma
    11·2 answers
  • Dres is taxed at a 17% tax rate for his federal taxes. Last year, he
    5·1 answer
  • Add: (7x-6y)+(3x+3y-3)
    13·2 answers
  • Dan says 3/5 is the same as 3.5 is he correct?
    7·2 answers
  • Bad gums may mean a bad heart. Researchers discovered that 79% of people who have suffered a heart attack had periodontal diseas
    15·1 answer
  • I need help on 13-22 PLEASE
    9·1 answer
  • Identify a and b for the hyperbola with equation
    14·1 answer
  • D(3, 6), E(-4,-3), F(6, 1) and<br> D'(1, 6), E'(8, -3), F(-2, 1)
    5·1 answer
  • Can y’all help me on this graph work asap pls
    11·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!