The main, and correct, method to finding the volume of a solid of revolution is by using the disc method, whereby you split the solid into numerous discs with a small thickness, say $\delta x$, and find the sum of the volumes of these discs. As $\delta x$ gets smaller, i.e. by taking the limit to 0, we calculate the volue of the whole solid of revolution.
This gives the following equation:
$Volume = \pi\int_a^b y^2 \delta x$
I understand how this works however am not sure why the following method doesn't work:
First; find the area under the curve of the function by simply doing $ Area = \int_a^b y \delta x$
Second; rotate this area by $360^o = 2\pi r$, where $r = y$ giving $Volume = 2\pi y\int_a^b y \delta x $
Ofcourse, this is not the correct equation to calculate the volume of a solid of revolution.
Why is this method wrong?