Chapter 7 Section 4 Exercise 5

Find the arc length of the function on the given interval.

\(f(x) = \frac{1}{3}x^{\frac{3}{2}}-x^{\frac{1}{2}}\) on \([0,1]\)

Define Function

func <- function(x) 1/3 * x^(3/2) - x^(1/2)

Plot

x <- seq(0,1,0.01)
plot(x,func(x), type="l",
     xlab="x", ylab="f(x)", main="f(x) on [0,1]")

Theoretical Solution

\(f'(x) = \frac{1}{3}\times \frac{3}{2}x^{\frac{1}{2}}-\frac{1}{2}x^{-\frac{1}{2}} = \frac{\sqrt{x}}{2} - \frac{1}{2\sqrt{x}} = \frac{x-1}{2\sqrt{x}}\)

\[ \begin{split} L_{arc} &= \int_0^1 \sqrt{1+f'(x)^2} dx =\\ &= \int_0^1 \sqrt{1+\frac{(x-1)^2}{4x}} dx = \\ &= \int_0^1 \sqrt{\frac{4x+x^2-2x+1}{4x}} dx = \\ &= \int_0^1 \sqrt{\frac{x^2+2x+1}{4x}} dx = \\ &= \int_0^1 \sqrt{\frac{(x+1)^2}{4x}} dx = \\ &= \int_0^1 \frac{x+1}{2\sqrt{x}} dx = \\ &= \frac{4}{3} \approx 1.333 \end{split} \]

R Solution

# Load pracma package
library(pracma)

# Convert function to parameterized function
F <- function(x) c(x, func(x))

# Find arc length
arclength(F, a=0, b=1)
## $length
## [1] 1.333331
## 
## $niter
## [1] 11
## 
## $rel.err
## [1] 3.825585e-06