Question

How do I find the volume generated by revolving a region enclosed by:

Answer 1

\[y=2x ^{2}\]
y=0
x=2

Answer 2

Around the x-axis?

Answer 3

Sorry, about the line y=8

Answer 4

Since you're rotating about the line y=8, you want everything in terms of x. This means that the function is \[y=x \sqrt{2}\](We don't lose anything because the region is always positive.)
Our limits of integration become x=0 (from y=0) and x=2 (from y=8). The equation for the area of a circle is A=pi*r^2. Our "height" (going horizontally now) will be a very small change in x, or dx. When we multiply an area by a height, we get a volume. This gives us the integral
\[\int\limits_{0}^{2}\pi*(x \sqrt{2})^2dx\].Now it's just a relatively simply integral.