Let's say I have a 2d-figure above the x-axis and I'm already given it's area (so I don't have to integrate). All of this figure touches the x-axis, so there's no hole in the middle when rotating
How would I calculate the volume of this figure if it's rotated around the x-axis.
So let's say the 2d-figure is a square sitting on the x-axis with an area of 16. How would I find the volume if this figure was rotating around the x-axis? Without setting up an integral of $4\pi$, because the figure may not always be a square whose equation we know.
Best Answer
There is a theorem called Pappus theorem which says the volume of revolution is the area of the region multiplied by the distance travelled by the center of mass of the object.
For example in case of a squareu with side $4$ and one side on the x axis the volume is $16\times 4\pi$ Thus we need the center of gravity to find the volume.