Final answer:
The center of mass for a rod with uniform density p = 2 is the midpoint of the interval, at x = 1. The center of mass for a rod with density p(x) = 3x³ + xe* over the interval (0, 2) requires integration, and the given value seems to correspond to the moment of inertia, not the center of mass.
Step-by-step explanation:
To compute the center of mass of a rod with a uniform density p = 2 over an interval € (-1,5), you can use the formula for the center of mass of a uniform rod, which is simply the midpoint of the interval. Since the rod has a constant density, the center of mass is at the midpoint (2/2), which is 1. Hence, the center of mass c is at x = 1.
For the second part, calculating the center of mass i on a rod where the density varies as p(x) = 3x³ + xe*, over the interval € (0, 2), requires integration. The center of mass is given by the formula:
I = ∫ x p(x) dx / ∫ p(x) dx
You mentioned that I = 12.5939, which seems to be the moment of inertia rather than the value required for the center of mass calculation. Assuming the correct calculation of the center of mass has been provided, we would round it to four decimal places to give the result.
However, if you need to compute the center of mass from the given density function, you'll have to perform the integrals mentioned above over the specified range, and then divide the results accordingly to obtain the center of mass.