- #1

- 137

- 6

## Main Question or Discussion Point

Hello all!

I'm just wanting a quick clarification on how finding the area under a polar graph works. Say we have the polar graph of ##r\left(\theta \right)=\frac{\arctan \left(2\theta \right)}{\theta }## as shown below:

I know that the area under the graph between ##0## and ##\frac{\pi }{2}## can be found with ##\int _0^{\frac{\pi }{2}}\left(\frac{1}{2}\left(\frac{\arctan \left(2\theta \right)}{\theta }\right)^2\right)d\theta ##. However, the thing I'm not quite sure about is if this integral finds only the red shaded area in the above image, or if it is both the red and blue shaded areas. Any help with this would be much appreciated :)

I'm just wanting a quick clarification on how finding the area under a polar graph works. Say we have the polar graph of ##r\left(\theta \right)=\frac{\arctan \left(2\theta \right)}{\theta }## as shown below:

I know that the area under the graph between ##0## and ##\frac{\pi }{2}## can be found with ##\int _0^{\frac{\pi }{2}}\left(\frac{1}{2}\left(\frac{\arctan \left(2\theta \right)}{\theta }\right)^2\right)d\theta ##. However, the thing I'm not quite sure about is if this integral finds only the red shaded area in the above image, or if it is both the red and blue shaded areas. Any help with this would be much appreciated :)