Variable Density So far we have considered cases where the density is constant. If we consider the case where the density changes as a function of x, we no longer can view the mass as the product of density and area. Instead f(Xk ) we can again cutType equation here. the region up into rectangles. To approximate the mass of each rectangle we will assume the density to be constant over the rectangle. Then the mass of the rectangle is approximately equal to the density at the center times the area of the rectangle mk = P(Xk) . f(Xk) . Ax XK-1 Xk XK Taking the limit of the sum of the masses of these rectangles we have n-+00 m = lim > P(Xx) . F(Xx) . Ax = [ P(xx) . f(x) dx Exercises 9) Modify the equations in (7) to find the moments of a lamina if the density, p(x), is a function of x. Mx = lim dx 1-+00 My = lim IM dx n-+00 10) Consider the lamina described in (8b). Suppose this lamina has a density given by p = cosx. Find the mass, moments, and center of mass of this lamina. Compare your answer to (8b). Do the answers make sense? Regions Between Two Curves For the remainder of the lab we will return to the case where the density is constant. We will conclude by considering what happens when the region lies between two curves y = f (x) and y = g(x), where f (x) 2 g(x). Since the density is constant we can use what we learned in Section 6.1 to conclude g(x) m = p [f (xx) - g(x)] dx