Hi could i please have help with this hw question.
"Show that the volume of a sphere is given by the formula V= 4/3 pi.r^3 by rotating the semicircle y= (r^2-x^2)^1/2 about the x axis.
Thanks
Sure! So that is a typical semicircle, radius \(r\), sitting above the \(x\) axis. Therefore, it crosses the \(x\) axis at \(-r\) and \(r\): Those are the bounds of our integral!
If you know the volume formula and how it works, this
should be reasonably clear on its own, but definitely let me know if it isn't
the only thing to note, besides the fact that we use the function
squared as usual, is that we are integrating with respect to \(x\) ONLY - We treat \(r\) as a constant!
We
also could have used symmetry to simplify the integral, by evaluating this instead:
You would get the same answer
let me know if any of those steps needs clarifying!