Preview Activity 11.4.1. Suppose that we have a flat, thin object (called a lamina) whose density varies across the object. We can think of the density on a lamina as a measure of mass per unit area. As an example, consider a circular plate DD of radius 1 cm centered at the origin whose density varies depending on the distance from its center so that the density in grams per square centimeter at point (x,y) is

(x,y)=102(x2+y2).

1. Suppose that we partition the plate into subrectangles Rij, where 1im1im and 1jn,1jn, of equal area A,A, and select a point (xij,yij) in Rij for each ii and j.j. What is the meaning of the quantity (xij,yij)A?

2. State a double Riemann sum that provides an approximation of the mass of the plate.

3. Explain why the double integral

   D(x,y)dA

4. tells us the exact mass of the plate.

5. Determine an iterated integral which, if evaluated, would give the exact mass of the plate. Do not actually evaluate the integral. (This integral is considerably easier to evaluate in polar coordinates, which we will learn more about in Section 11.5.)