What is the arc length of f(x) = x^2e^(3x) on x in [ 1,3] ?

1 Answer
Nov 15, 2016

f(x) = x^2e^(3x)

By the product rule:
f'(x) = (x^2)(d/dxe^(3x)) + (d/dxx^2) (e^(3x))
:. f'(x) = (x^2)(3e^(3x)) + (2x) (e^(3x))
:. f'(x) = 3x^2e^(3x) + 2x e^(3x)
:. f'(x) = (3x + 2)xe^(3x)

The Arc Length is given by:
S = int_1^3 sqrt(1 + f'(x)^2)
S = int_1^3 sqrt(1 + ((3x^2 + 2x)e^(3x))^2)
S = int_1^3 sqrt(1 + (3x^2 + 2x)^2e^(6x))

This definite integral does not have an intrinsic solution and would need to be solved numerically, using either a computer or estimated using the Trapezium Rule or Simpson's Rule