Consider the plate bounded by y2 = 8x and x = 2 with density 8 = 2-x. (a) Sketch a graph of the plate. Shade in the region and number the axes so that the important points on the plate can be read from the graph. (b) Express the mass of the plate as a double integral and evaluate to find the mass. (c) Express the x-coordinate of the center of mass of the plate as a double integral and evaluate to find the value of . As a side note, by the symmetry of the plate and density, the y-coordinate of the center of mass y = 0. It might be good practice to confirm this though.

Find a center of mass of a thin plate of density 8 = 5 bounded by the lines y = x and x = 0 and the parabola y = 6 - x² in the first quadrant.

A solid lies between two parallel planes 5 feet apart and has a volume of 45 cubic feet. What is the average area of cross- sections of the solid by planes that lie between the given ones?

The Deligne Dam on the Cayley River is built so that the wall facing the water is shaped like the region above the curve y=0.5x^2 and below the line y=260. (Here, distances are measured in meters.) The water level is 32 meters below the top of the dam. Find the force (in Newtons) exerted on the dam by water pressure.(Water has a density of 1000kg/m31000kg/m3, and the acceleration of gravity is 9.8m/sec2.9.8m/sec2. )