How to find the area of a circle by integration?

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How to find the area of a circle by integration?

Mathematics
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do you have an equation and what coordinates are you in?
say x^2 + y^2 = 1
\[Area = 2 \pi \int\limits_{0}^{R}r dr\] where R is radius of circle

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In doing so I am trying to find pi
haha prob not what you're looking for
\[x^{2} + y^{2} = 1\] \[y = \sqrt{1-x^{2}}\] this is top half of circle...integrate from -1 to 1 \[Area = 2\int\limits_{-1}^{1}\sqrt{1-x^{2}} dx = \pi\]
Yes, but how do i integrate \[\sqrt{1-x^2}\]
using trig substitution \[x = \sin u\] \[dx = \cos u\]
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
Yes, but why? And how come we can do that?
why can we use trig sub? you can substitute anything you want to make the integral more manageable

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