Integrating a Circle: Troubleshooting Simple Experiments

[tuoni]

Just started learning integration, and although I can manage simple stuff, I've run into problems with some experiments of mine.

http://enes.fi/temp/circle.png

The function for a circle is:

$y = \sqrt{r^{2} - x^{2}}$

; and Wolfram gives the integral as:

$\frac{1}{2}\left[x \cdot \sqrt{r^{2} - x^{2}} + r^{2} \cdot tan^{-1} \left(\frac{x}{\sqrt{r^{2} - x^{2}}}\right)\right]$

This in turn results in:

$f(b) - f(a) = f(s) - 0$

However, it's all wrong!

A radius of 10 means the area is approximately 78.540. Plugging in 10 into the equation results in division by zero. Even trying to integrate from 0 to 1, I get an area of 291.934, when it should be slightly less than 10!

Grrr! What am I doing wrong?

 

[ice109]

you have to take a limit as x -> r then you get the right answer

what you should do is do the integral in polar coordinates

 

[tuoni]

Thank you!

Setting my calculator to use rad (was set on deg) and using limits made it work.