## Volumes of solids of revolution

Determine the volume of

• #### Variant 1

a cylinder with height $$v$$ and a base radius $$r$$.

• #### Variant 2

a cone with height $$v$$ and base radius $$r$$.

• #### Variant 3

a sphere with radius $$r$$.

• #### Variant 4

a torus with radii $$R$$ and $$r$$, $$R > r$$.

• #### Variant 5

an infinite „funnel“ resulting form the rotation of $$f(x) = \frac{1}x$$ for $$x \in [1,\infty]$$ around the $$x$$ axis.

• #### Variant 6

a solid of revolution resulting from the rotation of the function $$y = \root 3 \of x$$ for $$y \in[1, 2]$$ around the $$y$$ axis.