How do you find the length of the curve for #y=x^2# for (0, 3)?

1 Answer
Steve M
Nov 3, 2016

Arc Length #= 1/4sinh^-1 6 + 3/2sqrt(37)#

Explanation:

The Arc Length #l# is given by the integration formula
# l=int_a^b sqrt(1+(dy/dx)^2)dx#

With #y=x^2 => dy/dx=2x#. And so:

# l = int_0^3 sqrt(1+(2x)^2)dx #
# :. l = int_0^3 sqrt(1+4x^2)dx #

I will quote the result, but if you want to see how to perform the integration, please use this link

# l = [sinh^-1(2x)/4 + (xsqrt(4x^2+1))/2]_0^3 #
# :. l = (sinh^-1 6/4 + (3sqrt(36+1))/2) - (sinh^-1 0/4 + 0)#
# :. l = (sinh^-1 6/4 + (3sqrt(37))/2) - (0 + 0)#
# :. l = 1/4sinh^-1 6 + 3/2sqrt(37)#

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