7.2 Problem 7

Find the volume between the \(y=\sqrt{x}\) and \(y=x\) rotated about the x axis in the domain (0,1)

The area function will be \(\pi{r}^{2}\) for each curve will be, so we just need to multiply our original function by \(\pi\) and square it.

\(\pi ( \int _{ 0 }^{ 1 }{ xdx } - \int _{ 0 }^{ 1 }{ { x }^{ 2 }dx) }\)

Using R for the integration:

func1 <- function(x){
return (pi*x)
}

func2 <- function(x){
  return(pi*x**2)
}

integrate(func1, 0, 1)$value - integrate(func2, 0, 1)$value
## [1] 0.5235988

Those inegrals will work out to \(\frac{1}{2}\) and \(\frac{1}{3}\), so the expression evaluates to \(\frac{1}{6}\pi\) which is consistent with the answer from R