Final answer:
The length L of the arc defined by the function y = e^(4x) / 4 from x=0 to x=3 is found by integrating the square root of 1 plus the square of the function's derivative from 0 to 3.
Step-by-step explanation:
To find the length L of the arc described by the function y = e^(4x) / 4, for the interval 0 ≤ x ≤ 3, we use the formula for the length of a curve in the plane defined by a function y = f(x):
L = ∫_{a}^{b} √{1 + [f'(x)]^2} dx
First, we find the derivative f'(x) of our function f(x) = e^(4x) / 4, which is:
f'(x) = e^(4x)
We square this derivative and add 1 to find the integrand for our arc length:
(f'(x))^2 = (e^(4x))^2 = e^(8x)1 + (f'(x))^2 = 1 + e^(8x)
Then, we integrate the square root of this quantity from the limits of integration given, which are 0 and 3:
L = ∫_{0}^{3} √{1 + e^(8x)} dx
Since this integral may be complex and may not have a simple antiderivative, it is suggested to use a Table of Integrals or computational tools to evaluate the integral and find the arc length L.