1. ## Integral

I want to evaluate the volume inside $r=cos(\theta)$.
Thanks.

2. \begin{aligned}r & = \cos \theta \\ r^2 & = r\cos \theta \\ x^2 + y^2 & = x \\ (x-\tfrac{1}{2})^2 + y^2 & = \tfrac{1}{4} \end{aligned}

Revolving it around the x-axis produces a sphere with radius $\tfrac{1}{2}$ .

3. 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.

$2{\pi}\int_{0}^{\frac{1}{2}}(\frac{1}{4}-x^{2})dx$

or leave it where it is and:

${\pi}\int_{0}^{1}\left(\frac{1}{4}-(x-\frac{1}{2})^{2}\right)dx$

4. I want the volume within $\sqrt{x^2+y^2+z^2} = \frac{x}{\sqrt{x^2+y^2}}$*
Here is a picture of the solid.

5. I want the volume within *
I suggest you check that you have the right definition of $\theta$ because my book used it for the other angle. Anyway, using your notation:
$\int\int\int dV=\int_0^{\pi}\int_0^{2\pi}\int_0^{cos(\theta)}r^ 2\sin(\phi)dr d\theta d\phi$

6. Physicists and mathematicians use the opposite definition right.
If you use the physicists' you get a sphere.
If you the mathematicians' you get this weird figure I've showed above.