Final answer:
To find the length of the curve y = x * e^(9x), we used the arc length formula and derived the integral L = ∫_{0}^{3} √[1 + (e^(9x) + 9x * e^(9x))^2] dx without evaluating it.
Step-by-step explanation:
To set up an integral for the length of the curve given by y = x * e^(9x) from x = 0 to x = 3, we will use the arc length formula for a function y = f(x), which is:
L = ∫_{a}^{b} √[1 + (dy/dx)^2] dx
To apply this formula, we need to find the derivative dy/dx. The derivative of y = x * e^(9x) with respect to x is obtained using the product rule:
dy/dx = e^(9x) + 9x * e^(9x)
Then the square of the derivative is:
(dy/dx)^2 = (e^(9x) + 9x * e^(9x))^2
Now the integral for the arc length L from x=0 to x=3 can be set up as follows:
L = ∫_{0}^{3} √[1 + (e^(9x) + 9x * e^(9x))^2] dx
This integral represents the length of the curve y = x * e^(9x) over the interval [0, 3], but as the question requests, it has not been evaluated.