What is the arc length of #f(x)= sqrt(x^3+5) # on #x in [0,2]#?

1 Answer
Alvin L.
Dec 28, 2017

The length is around #2.58#.

Explanation:

The formula for arc length of a curve of a function #f(x)# on the interval #[a,b]# is equal to:
#int_a^bsqrt(1+(f'(x))^2)\ dx#

Let us first work out #f'(x)#:
#d/dx(sqrt(x^3+5))=#

I will use the chain rule and let #u=x^3+5#
#=d/(du)(sqrtu)*d/dx(x^3+5)=1/(2sqrtu)*3x^2=#

#=(3x^2)/(2sqrt(x^3+5))#

Now we plug this into our arc length formula:
#int_0^2sqrt(1+((3x^2)/(2sqrt(x^3+5)))^2)\ dx#

#int_0^2sqrt(1+(9x^4)/(4(x^3+5)))\ dx#

This function doesn't have an elementary anti-derivative (that I've been able to find). The best we can really do is use an approximation. Here I did a midpoint Riemann rectangle sum with 25 rectangles to get an answer of about #2.58#:
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