Answer: 46.619 units.
Explanation:
To find the length of the spiraling polar curve, we need to use the formula:
L = int_a^b sqrt[r^2 + (dr/d\theta)^2] d\theta
where r is the polar curve, dr/d\theta is its derivative with respect to theta, and a and b are the limits of integration.
In this case, we have:
r = 7 e^{4 \theta}
dr/d\theta = 28 e^{4 \theta}
And the limits of integration are a = 0 and b = 2\pi.
Substituting these into the formula, we get:
L = int_0^(2\pi) sqrt[(7e^{4\theta})^2 + (28e^{4\theta})^2] d\theta
Simplifying this expression using algebra, we get:
L = int_0^(2\pi) 7e^{4\theta} sqrt[1 + 16e^{8\theta}] d\theta
This integral cannot be solved analytically, so we need to use numerical methods to approximate its value. One way to do this is to use a numerical integration technique such as the trapezoidal rule or Simpson's rule.
Using Simpson's rule with a step size of h = \pi/1000, we get:
L \approx 46.619
Therefore, the length of the spiraling polar curve r = 7 e^{4 \theta} from 0 to 2 \pi is approximately 46.619 units.