# Find the volume generated by revolving the given region around $y$ axis.

Find the volume of solid generated by revolving the described region about $$y$$ axis

The region enclosed by the line $$y =x$$, $$y =-x/2$$ and $$x =2$$

So, this is how I setup the integral

Volume = $$\displaystyle 2\pi \int_{0}^{2}(2-x) (3x/2) dx$$

When I calculate this the answer comes out to be $$4\pi$$

However, the answer given to me is $$8\pi$$

Can anyone please check and tell me what is wrong with my solution ?

Thank you.

If you draw the figure in the $$xy$$ plane, you have a triangle with a vertex at the origin. Then, using cylindrical shells with radius $$x$$ and height $$3x/2$$, your integral is $$V=2\pi\int_0^2x\frac32 x dx=8\pi$$