Help with a revolution solid

Question

Suppose I have an area in the cartesian system formed by the $y$ axis and a given function $y=f(x)$. How do I evaluate the volume of the solid formed by completely revolving this area around the $y$ axis (given a known interval $x\in[a,b]$)? Assuming that you can't invert the function to a form like $x=f(y)$. Thank you in advance.

Answer 1

As described in the Wikipedia Solid of revolution article, the volume of the solid formed by rotating the area between the $y$ axis and $y = f(x)$ and lines $x = a$ and $x = b$ is $$V = 2 \pi \int_a^b x \lvert f(x) \rvert dx$$

For $f(x) \ge 0$, this is called shell integration, $$V = 2 \pi \int_a^b x \; f(x) \; dx$$