apple_pi Group Title How to find the area of a circle by integration? one year ago one year ago

1. juantweaver Group Title

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

2. apple_pi Group Title

say x^2 + y^2 = 1

3. dumbcow Group Title

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

4. apple_pi Group Title

In doing so I am trying to find pi

5. dumbcow Group Title

haha prob not what you're looking for

6. dumbcow Group Title

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

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

8. dumbcow Group Title

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

9. dumbcow Group Title

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 Group Title

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

11. dumbcow Group Title

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