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Determine the surface of a cardioid

  1. Jan 15, 2017 #1

    Math_QED

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    1. The problem statement, all variables and given/known data

    Consider the cardioid given by the equations:

    ##x = a(2\cos{t} - \cos{2t})##
    ##y = a(2\sin{t} - \sin{2t})##

    I have to find the surface that the cardioid circumscribes, however, I don't know what limits for ##t## I have to take to integrate over. How can I know that, as I don't know how this shape looks like (or more precisely where it is located)?

    2. Relevant equations

    Integration formulas

    3. The attempt at a solution

    I know how I have to solve the problem once I have the integral bounds, but I don't know how I have to determine these. In similar problems, we could always eliminate cost and sint by using the identity ##cos^2 x + sin^2 x = 1## but neither this nor another way to eliminate the cos, sin seems to work. This makes me think, would it be sufficient if I find the maximum and minimum x-coordinate in function of t using derivatives? Then I would have bounds to integrate over, but this seems like a lot of work.
     
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  3. Jan 15, 2017 #2

    BvU

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    That's easy ! calculate a few points and connect the dots ! And when things start to repeat, you know you've passed the bound for ##t## :smile:
     
  4. Jan 15, 2017 #3

    Math_QED

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    I have to know what the bound exactly is, so I can use it as my integration bounds.
     
  5. Jan 15, 2017 #4

    BvU

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    O ?
    But you already have an exact expression for the bound :rolleyes:
     
  6. Jan 15, 2017 #5

    Math_QED

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    I don't think I understand what you mean. Can you elaborate?
     
  7. Jan 15, 2017 #6

    BvU

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    You wrote down the equations in post #1. They exactly define the bound. A bit corny of me, sorry.
    Did you make the sketch? Find the domain of ##t## ?

    What would your integral look like ?

    The section in your textbook/curriculum where this exercise appears: what's it about ?
     
  8. Jan 15, 2017 #7

    Math_QED

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    It's about integration: definite and indefinite integrals. The next chapter deals with arc length.
     
  9. Jan 15, 2017 #8

    BvU

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    That answers the last of my three questions. What about the other two ?
    let me add another one: do you know two ways to integrate to get the area of a circle ?

    I am a bit evasive because I am convinced you will say 'of course' once you have found your way out of this on your own steam....
     
  10. Jan 15, 2017 #9

    Math_QED

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    Yes I can find the integral of ##\sqrt{r^2 - x^2}## so that wouldn't be the problem. It's late now but I will try the problem tomorrow again and answer your questions then.
     
  11. Jan 15, 2017 #10

    BvU

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    That's one way. There's a quicker way too. Same way works for the cardioid (but it's admittedly less simple than for a circle -- still an easy integral).

    First three (not two) questions in post # 6 are still open ...

    And it's ruddy late here too.. :sleep:
     
  12. Jan 15, 2017 #11

    Math_QED

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    Yes I believe you can do x = cos t, y = sin t and then integrate over the correct bounds. I will answer the other questions tomorrow.
     
  13. Jan 17, 2017 #12

    BvU

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    get a good sleep ?
    Isn't good for an area (only one integrand), but it's in the right direction. You need a factor r in both.

    ##
    \sqrt{r^2 - x^2}\ ## are vertical strips, and I'm trying to lure you towards investigating pie pieces (sectors, so to say). As you picked up correctly.

    What would the integral for the cardioid area look like with this approach ?

     
  14. Jan 17, 2017 #13

    Math_QED

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    I figured out how to solve the problem using your hints. Thanks a lot.
     
  15. Jan 17, 2017 #14

    BvU

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    You're welcome.
     
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