Is it possible to obtain volume of a sphere by rotating two-dimensional circle 180 degrees (or half circle 360 degrees)? In instances like calculating volume of cubes and 3d rectangular objects, volume is obtained by multiplying 2d surface by another scalar in the 3rd dimension.
By the same logic, a circle is a 2d entity and its rotation takes place in 3rd dimension. I am not sure what specific method would be suitable but general "intuition" makes sense.
$\endgroup$ 22 Answers
$\begingroup$It is possible but, respect to simple translation for cubes or parallelepipedes, we need to "adjust" the evaluation to take into account that the solid is obtained by a rotation with respect to an axes, notably it can be done by Pappus's centroid theorem which states that
the volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by its geometric centroid
Sure, rotate the function $$ f(x)=\sqrt{r^2-(x-r)^2} $$ about the $x$ axis to find $$ \pi\int_0^{2r} f(x)^2\mathrm dx=\pi\left( 2r^3-\frac{2r^3}{3} \right)=\pi\frac{4r^3}{3} $$ Noting the geometric intuition here: we are summing up the areas of circles (we consider them flat, since their width is an infinitesimal $\mathrm dr$) sweeping out of the page.
$\endgroup$ 3
