Mass worksheet Learning objectives: e Apply the formula mass = (density)(volume) to derive the method for finding mass of a lamina with variable density e Find the mass of a lamina with variable density Definition For our purposes, a lamina is a thin plate (of negligible thickness) occupying a region in the xy-plane. 1. Suppose you want to use a double integral fi region D R*. What would you use for f(x, y)? f(x, y)dA to compute the area of a 2. Use a double integral to find the area of the lamina that occupies the region bounded by y=1,x =1,andx+y=1inR?. If a lamina has uniform (constant) density, then the mass of the lamina can be found using the formula mass = (density)(area). 3. Suppose the lamina described above has density p = 3 kg/m4*2. What is the mass of the lamina? If the lamina has variable density, then we can't directly use the formula mass = (density)(area). Instead, we should imagine breaking the problem up into small pieces and approximating. Consider a lamina occupying a region R in the xy-plane, with variable density given by p(x, y). Suppose you look at a little rectangular piece of that lamina, whose width is Ax and whose length is Ay. 4. What is the approximate mass of that little piece? 5. What will you do in order to approximate the mass of the entire lamina? 6. Then, what will you do to get a better approximation? 7. Finally, what will you do to find the exact mass of the lamina? Thus, the mass of a lamina occupying the region R in the xy-plane with variable density given by p(x, y) is I, p(x, y)dA. 8. Find the mass of the lamina that occupies the region bounded by y = 1,x = 1,andx + y = 1 in R? that has variable density given by p(x,y) =x + y. 9. Find the mass of the lamina that occupies the region bounded by y = e~, y = 0,x = 0, and x = 1, R? that has variable density given by p(x, y) = y. 10. Find the mass of the lamina that occupies part of the disk x* + y*