How do you calculate the arc length of the curve y=x^2 from x=0 to x=4?

1 Answer
May 16, 2018

Use the arc length formula.

Explanation:

y=x^2

y'=2x

Arc length is given by:

L=int_0^4sqrt(1+4x^2)dx

Apply the substitution 2x=tantheta:

L=1/2intsec^3thetad theta

This is a known integral:

L=1/4[secthetatantheta+ln|sectheta+tantheta|]

Reverse the substitution:

L=1/4[2xsqrt(1+4x^2)+ln|2x+sqrt(1+4x^2)|]_0^4

Hence

L=2sqrt65+1/4ln(8+sqrt65)