What is the arc length of f(x)= sqrt(x^3+5) f(x)=√x3+5 on x in [0,2]x∈[0,2]?

1 Answer
Alvin L.
Dec 28, 2017

The length is around 2.582.58.

Explanation:

The formula for arc length of a curve of a function f(x)f(x) on the interval [a,b][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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