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 \(D\) of radius 1 cm centered at the origin whose density \(\delta\) varies depending on the distance from its center so that the density in grams per square centimeter at point \((x, y)\) is
Suppose that we partition the plate into subrectangles \(R_{ij}\text{,}\) where \(1 \leq i \leq m\) and \(1 \leq j \leq n\text{,}\) of equal area \(\Delta A\text{,}\) and select a point \((x_{ij}^*,y_{ij}^*)\) in \(R_{ij}\) for each \(i\) and \(j\text{.}\) Write a couple of sentences to explain the meaning of the quantity \(\delta(x_{ij}^*,y_{ij}^*) \Delta A\text{.}\)
State 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 12.5.)
