## apple_pi 3 years ago How to find the area of a circle by integration?

1. juantweaver

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

2. apple_pi

say x^2 + y^2 = 1

3. dumbcow

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

4. apple_pi

In doing so I am trying to find pi

5. dumbcow

haha prob not what you're looking for

6. dumbcow

$x^{2} + y^{2} = 1$ $y = \sqrt{1-x^{2}}$ 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

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

8. dumbcow

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

9. dumbcow

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

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

11. dumbcow

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