I want to evaluate the volume inside $\displaystyle r=cos(\theta) $.
Thanks.
That is just a circle of radius 1 centered at (1/2,0)
A sphere if you revolve
If you must do this with integration, you can just move it over the the origin.
Now, revolve a circle of radius 1/2.
$\displaystyle 2{\pi}\int_{0}^{\frac{1}{2}}(\frac{1}{4}-x^{2})dx$
or leave it where it is and:
$\displaystyle {\pi}\int_{0}^{1}\left(\frac{1}{4}-(x-\frac{1}{2})^{2}\right)dx$