Question

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

Answer 1

`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