Final answer:
To find the center of mass of a rod with a varying density, you need to integrate the product of the density function and the position function over the interval. In this case, the density function is given as p(x) = 5^3 + e^-x and the interval is (0, 1).
Step-by-step explanation:
The center of mass of a rod with a varying density can be found by integrating the product of the density function and the position function over the interval of interest. In this case, we have the density function p(x) = 5^3 + e^-x, and we want to find the center of mass in the interval (0, 1).
To do this, we first calculate the mass of the rod by integrating the density function over the interval:
m = ∫[0, 1] (5^3 + e^-x) dx
Next, we calculate the moment of mass about the origin by integrating the product of the density function and the position function over the interval:
M_0 = ∫[0, 1] x(5^3 + e^-x) dx
Finally, we divide the moment of mass by the total mass to find the center of mass:
x_c = M_0 / m