Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.
$$y = x − x^2, y = 0; about x = 2$$
Can't rewrite the equation so it is integrable in terms of the height (y). Tried a variety of strategies, none seem to be panning out. Anyone have an idea on how to solve this problem?
Intuitive explanation:
The idea is that you take hollow cylinders of thickness $\Delta x$ and so the cylindrical shell's volume would then be $V=\text{height}\cdot\text{radius}\cdot\Delta x$. If you then let $\Delta x\to0=dx$ you get the integration formula. $$V=\int_a^b2\pi(h-x)f(x)dx\text{ if }a<b\leq h$$ $$V=\int_a^b2\pi(x-h)f(x)dx\text{ if }h\leq a<b$$