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Math Help - [SOLVED] Finding the area inside a loop?

  1. #1
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    [SOLVED] Finding the area inside a loop?

    The problem says that the curve makes a loop along the x-axis, and asks for the area inside the loop. How do I go about finding this? This seems slightly different than the other area problems we have been given.

    x=25-t^2
    y=t^3-25t

    dx/dt=-2t
    dy/dt=3t^2-25
    dy/dx=3t^2-25/-2t

    But what do I do from here?
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  2. #2
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    Anyone know how to solve this problem? I am stuck from this point.
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  3. #3
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    Yea I do, I just thought my method was a bit tacky. I mean, how do you solve for the area under a function using the parametric expressions? Is that really easy and I'm suppose to know that already or what (I'm older than you and can't use that excuse)? See what I mean. Ok, so I just wrote it in terms of y(x)=\pm x\sqrt{25-x}. Now look at the plot below:

    For large negative t, both x and y are negative. When t hits -5, they're zero, into the first quad, looping down, reaching the x-axis when t=0 right? Then it goes into the 4th quad, back to the origin at t=5 and then into the 2nd quad. So I'd solve for y in terms of x:

    y(x)=\pm x\sqrt{25-x}

    We can integrate just the top part since it's reflective and I'll use the positive square root:

    2\int_0^{25} x\sqrt{25-x}dx
    Attached Thumbnails Attached Thumbnails [SOLVED] Finding the area inside a loop?-loop2.jpg  
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