Exercise 18

A triangular plate is bounded by the graphs of the equations $y = 2x$, $y = 4x$, and $y = 4$. The plate's density at $(x,y)$ is given by $\delta(x,y) = 4xy^2 + 1$, measured in grams per square centimeter (and $x$ and $y$ are measured in centimeters).

1. Set up an iterated integral whose value is the mass of the plate. Include a labeled sketch of the region of integration. Why did you choose the order of integration you did?

2. Determine the mass of the plate.

3. Determine the exact center of mass of the plate. Draw and label the point you find on your sketch from (a).

4. What is the average density of the plate? Include units on your answer.