bevis vha. diff. regning at rumfanget for en kugle er 4/3*π*r^3

se https://www.studieportalen.dk/Forums/Thread.aspx?id=1233043

       drej halvcirklen
                                          y = √(r² - x²)              -r < x < r

                                                                                                    360º om x-aksen

      og beregn drejningslegemets rumfang

Det er så snarere integralregning, der benyttes.

                        V_x  = π · _-r ∫ ^r (f(x))²dx

 V_x  = π · _-r ∫ ^r y²dx = π · _-r ∫ ^r (r² - x²)dx  = π· [r²·x - (1/3)·x³]_-r^r   =   π·(r²·r - (1/3)·r³ - (r²·(-r) - (1/3)·(-r)³)) =

                                               π·(r³ - (1/3)·r³ + r³ - (1/3)·r³) = π·(1 - (1/3)  + 1 - (1/3) )·r³ =

                                               π·((6/3) - (1/3)  - (1/3) )·r³ = (4/3)·π·r³