How to find the area of a circle by integration?

Question

anonymous · Answer

do you have an equation and what coordinates are you in?

anonymous · Answer

say x^2 + y^2 = 1

dumbcow · Answer

$$Area = 2 \pi \int\limits_{0}^{R}r  dr$$
where R is radius of circle

anonymous · Answer

In doing so I am trying to find pi

dumbcow · Answer

haha prob not what you're looking for

dumbcow · Answer

$$x^{2} + y^{2} = 1$$
$$y = \sqrt{1-x^{2}}$$
this is top half of circle...integrate from -1 to 1
$$Area = 2\int\limits_{-1}^{1}\sqrt{1-x^{2}} dx = \pi$$

anonymous · Answer

Yes, but how do i integrate $$\sqrt{1-x^2}$$

dumbcow · Answer

using trig substitution
$$x = \sin u$$
$$dx = \cos u$$

dumbcow · Answer

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

anonymous · Answer

Yes, but why? And how come we can do that?

dumbcow · Answer

why can we use trig sub?
you can substitute anything you want to make the integral more manageable