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Math Help - find the area

  1. #1
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    find the area

    I neeeed help ... important

    SCHAUM`S OUTLINES
    THEORY AND PROBLEMS OF DIFFERENTIAL AND INTEGRAL
    CHAPTER 34 - PLANE AREAS BY INTEGRAL
    PROBLEM 15 -T


    here .... the picture

    find the area bounded by the given curve

    Attached Thumbnails Attached Thumbnails find the area-untitled.gif  
    Last edited by rqeeb; March 27th 2010 at 05:45 AM.
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  2. #2
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    A = \int_0^\pi \left(x(t)^2 + y(t)^2\right) \, dt


    note that this cardioid can also be represented by the equation

    r = 2-2\cos{t}

    and the area of this cardioid is the same ...

    A = \int_0^\pi r(t)^2 \, dt

    ... which leads me to believe that if you take the expressions for x(t) and y(t), square each, and find their sum, you'll get r^2.

    I'll leave it for you to complete the necessary algebra/trig grunt work for confirmation.
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  3. #3
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    thank you sir
    I complete what you mention in the picture .. and I get the correct answer 6pi ,but to tell you the truth I haven`t study this way by using polar coordinates (if the teacher ask me why from 0 to pi !!! I have no idea).


    but if you don`t mind I`ll show you what I did and give me your help ... 30 minutes later.


    I am sorry about my English ..It isn`t my native language ... I am studying it
    Attached Thumbnails Attached Thumbnails find the area-untitled.gif  
    Last edited by rqeeb; March 27th 2010 at 07:36 AM.
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  4. #4
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    let me start with the black area in the previos picture

    could you please correct what I did
    Is it right or wrong?
    Attached Thumbnails Attached Thumbnails find the area-untitledd.gif  
    Last edited by rqeeb; March 27th 2010 at 07:37 AM.
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  5. #5
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    Quote Originally Posted by rqeeb View Post
    thank you sir
    I complete what you mention in the picture .. and I get the correct answer 6pi ,but to tell you the truth I haven`t study this way by using polar coordinates (if the teacher ask me why from 0 to pi !!! I have no idea).


    A = \int_0^{2\pi} \frac{r^2}{2} \, dt

    using symmetry (upper half of the cardioid has the same area as the lower half), so integrate from 0 to \pi and double the area ...

    A = 2\int_0^{\pi} \frac{r^2}{2} \, dt = \int_0^{\pi} r^2 \, dt
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