# Some of the pioneers of calculus, such as Kepler and Newton, were inspired bythe problem of finding the volumes of wine barrels. (In fact Kepler published abook Stereometria doliorum in 1615...

Some of the pioneers of calculus, such as Kepler and Newton, were inspired by
the problem of finding the volumes of wine barrels. (In fact Kepler published a
book Stereometria doliorum in 1615 devoted to methods for finding the volumes of barrels.) They often approximated the shape of the sides by parabolas.
(a) A barrel with height h and maximum radius R is constructed by rotating about the x-axis the parabola y = R-cx^2, -h/2`<=` x `<=`  h/2, where c is a positive constant. Show that the radius of each end of the barrel is r = R-d, where d = (ch^2)/4.

aruv | Student

Given curve  `y=R-cx^2` , which is parabola and open down.

Let us write this parabola as

`x^2=(R-y)/c`

`=>`  Parabola is defined if `R-y>=0`

`=>R>=y`

From above equation , we have

`x=0 => y=R`  and `y=0=>x=+-sqrt(R/c)`

A barrel with height h and maximum radius R is constructed by rotating about the x-axis the parabola y = R-cx^2, -h/2 x  h/2, where c is a positive constant.

`=> `  If  `x=h/2`  then the value of y will be the radius of the barrel.

`=> (h/2)^2=(R-y)/c`

`=>(ch^2)/4=R-y`

`=>y=R-(ch^2)/4`

Thus the radius of the barrel  `r=y=R-(ch^2)/4`

`r=R-d ,` where `d=(ch^2)/4` .