1. Find 0∫10∫1-x0∫1-x-y z dzdydx.
2. A sheet of metal in the shape of a triangle massing 10 kg per square meter is to be spun at an angular velocity of 4 radians per second about some axis perpendicular to the plane of the sheet. The triangle is a right triangle with both short sides of length 1 meter.
(a) The axis of rotation is the line through the right angle of the triangle and perpendicular to the plane of the sheet. What is the resulting kinetic energy?
(b) The axis of rotation is instead through the centroid of the triangle, or equivalently, through the center of mass of the sheet. That center of mass is two thirds of the way from any vertex to the midpoint of the side opposite it. Set up, but do not evaluate, a double integral which if evaluated would give the kinetic energy of the spinning plate. (The formula for kinetic energy is (1/2)mv2 where m denotes mass and v denotes regular speed, which is not the same thing as angular velocity.)
3. Johnny has a checkerboard and he put checkers on each of the squares of the checkerboard, each right in the middle. For each of the checkers, he computed its x and y coordinates, counting the distance all the way across the checkerboard as 1, so that each square had sides of length 1/8. He took for his axes the bottom left corner of the board, so that the first checkerAc€?cs coordinates were (1/16, 1/16), and the next one over had coordinates (3/16, 1/16), and so on. For each of the sixty four checkers on the board, he computed (x2+y2)(x+y). He added up all sixty four numbers he got. By relating Riemann sums to double integrals, but WITHOUT doing all that arithmetic, estimate the sum he got.