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

1 Answer
Steve M
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

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