What is the arc length of #f(x)=x^2/sqrt(7-x^2)# on #x in [0,1]#?

Question

Emma Aldridge · Accepted Answer

The arc length of a function #f# on the interval #[a,b]# is given by:

#s=int_a^bsqrt(1+(f'(x))^2)dx#

Here, #f(x)=x^2/sqrt(7-x^2)# and I leave it to you to determine that #f'(x)=(x(14-x^2))/(7-x^2)^(3/2)#.

Then, the arc length desired is:

#s=int_0^1sqrt(1+(x^2(14-x^2)^2)/(7-x^2)^3)dx#

This has no closed form. Use a calculator to find:

#sapprox1.10458#

Emily Ammerman · Answer

#approx 1.10458# We have #f(x)=x^2/sqrt(7-x^2)# so #f'(x)=(2x*sqrt(7-x^2)-x^2*1/2*(7-x^2)^(-1/2)(-2x))/(7-x^2)# Multiplying numerator and denominator by #sqrt(7-x^2)# we get #f'(x)=(2x(7-x^2)+x^3)/((7-x^2)*sqrt(7-x^2))# so #f'(x)=(14x-x^3)/((sqrt(7-x^2)(7-x^2))# Using the Formula #s=int_0^1sqrt(1+f'(x)^2)dx# we get by a numerical method #approx 1.10458#

William Abdalla · Answer

undefined To find the arc length of the curve $f(x) = \frac{x^2}{\sqrt{7 - x^2}}$ on the interval $[0, 1]$, you can use the formula for arc length, which is given by:

$$ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx $$

First, find the derivative $ \frac{dy}{dx} $ of $f(x)$, then plug it into the formula along with the limits of integration $[0, 1]$, and integrate to get the arc length.