A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

anonymous

  • one year ago

What integral represents the volume of the solid formed by revolving the region bounded by the graphs of y = x^3 y = 1 and x = 2 about the line x = 2

  • This Question is Closed
  1. mLe
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Hello @Yaros and Welcome to OpenStudy! Thanks for asking a Qualified Help Question first try graphing out what you have been given so that you have a visual representation of what you are looking for :)

  2. Michele_Laino
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    your problem can be represented by this drawing: |dw:1435594893861:dw|

  3. Michele_Laino
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    now I make this traslation: \[\Large \left\{ \begin{gathered} x = X + 2 \hfill \\ y = Y \hfill \\ \end{gathered} \right.\] where X, Y are the new coordinates, furthermore, we can see that the origin of the plane X-Y is lòocated at point x=2,y=0

  4. Michele_Laino
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    |dw:1435595055446:dw|

  5. Michele_Laino
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    Now I refer the equation of our cubic parabola to the new reference system, so I get: \[\Large y = {x^3} \Rightarrow Y = {\left( {X - 2} \right)^3}\]

  6. Michele_Laino
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    so the requested volume is given by the subsequent integral: \[\Large V = \pi \int_0^1 {dY\left\{ {{{\left( {\sqrt[3]{Y} - 2} \right)}^2} - {{\left( { - 1} \right)}^2}} \right\}} \]

  7. Michele_Laino
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    oops.. the right equation of our paraboa using the coordinates X,Y is: \[\Large y = {x^3} \Rightarrow Y = {\left( {X + 2} \right)^3}\]

  8. Michele_Laino
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    |dw:1435595559191:dw|

  9. Michele_Laino
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    please keep in mind that we are considering the rotation, around the x=2 axis

  10. Michele_Laino
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    please keep in mind that we are considering the rotation, around the x=2 axis

  11. Michele_Laino
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    after a simple computation, we get: \[\Large V = \pi \left. {\left( {\frac{3}{5}{Y^{5/3}} + 3Y - 3{Y^{4/3}}} \right)} \right|_0^1 = \frac{{3\pi }}{5}\]

  12. Michele_Laino
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    |dw:1435596051708:dw|

  13. Michele_Laino
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    |dw:1435596146656:dw|

  14. Michele_Laino
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    that integral is the volume generated by the rotation of the shaded region above

  15. Michele_Laino
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    now we have to add the volume of the cylinder whose radius is r=1 and height =1

  16. Michele_Laino
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    so we get: \[\Large {V_{TOTAL}} = V + \pi = \frac{{3\pi }}{5} + \pi = \frac{{8\pi }}{5}\]

  17. Michele_Laino
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    Another way to compute the requested volume, is to apply the subsequent formula: \[\Large {V_{TOTAL}} = \pi {\int_0^1 {dY\left( {\sqrt[3]{Y} - 2} \right)} ^2}\]

  18. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.