Q. Find the volume of wedge intercepted between the cylinder $x^2+y^2=2ax$ and the planes $z=x$ and $z=2x$
I am not able to make out how to find its volume using cylindrical co-ordinates in this case. I know the basics of cylindrical co-ordinates but basically not able to apply it to find limits of the integral. some tips on that would be very helpful
thanks
The equation of the circle in polar coordinates is $r=a \cos{\theta}$. Thus we may set up the integral in cylindricals as
$$\int_{-\pi/2}^{\pi/2} d\theta \, \int_0^{2 a \cos{\theta}} dr \, r \, \int_{r \cos{\theta}}^{2 r \cos{\theta}} dz $$
This ends up simplifying to
$$\frac{16 a^3}{3} \int_{-\pi/2}^{\pi/2} d\theta \, \cos^4{\theta}$$
I get $\pi a^3$ as the volume.