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apple_pi

  • 3 years ago

How to find the area of a circle by integration?

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  1. juantweaver
    • 3 years ago
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    do you have an equation and what coordinates are you in?

  2. apple_pi
    • 3 years ago
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    say x^2 + y^2 = 1

  3. dumbcow
    • 3 years ago
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    \[Area = 2 \pi \int\limits_{0}^{R}r dr\] where R is radius of circle

  4. apple_pi
    • 3 years ago
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    In doing so I am trying to find pi

  5. dumbcow
    • 3 years ago
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    haha prob not what you're looking for

  6. dumbcow
    • 3 years ago
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    \[x^{2} + y^{2} = 1\] \[y = \sqrt{1-x^{2}}\] this is top half of circle...integrate from -1 to 1 \[Area = 2\int\limits_{-1}^{1}\sqrt{1-x^{2}} dx = \pi\]

  7. apple_pi
    • 3 years ago
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    Yes, but how do i integrate \[\sqrt{1-x^2}\]

  8. dumbcow
    • 3 years ago
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    using trig substitution \[x = \sin u\] \[dx = \cos u\]

  9. dumbcow
    • 3 years ago
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    in the end you will just get pi =pi if you want to use this to get numerical approximation for pi, then integrate by computing area under curve using trapezoid rule or simpsons rule or something like that

  10. apple_pi
    • 3 years ago
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    Yes, but why? And how come we can do that?

  11. dumbcow
    • 3 years ago
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    why can we use trig sub? you can substitute anything you want to make the integral more manageable

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