# Volumes by Cylindrical Shells

skip A but do B! times 7.1 Areas between Curves 7.1 Exercises 7.2 Volumes 7.2 Exercises 7.3 Volumes by Cylindrical Shells 7.3 Exercises 7.4 Arc Length The Arc Length Function 7.4 Exercises 7.5 Area of a Surface of Revolution 7.5 Exercises 7.6 Applications to Physics and Engineering Work Hydrostatic Pressure and Force Moments and Centers of Mass 7.6 Exercises 7.7 Differential Equations Separable Equations Logistic Growth Mixing Problems Direction Fields One problem is bonus. Duration: 50 min. Be neat, show your work to receive credit. 1 (5 points) The tank shown below is full of water. Set up (do not compute!) a definite integral representing the work required to pump the water out of the spout. Note that the density of water is 1000 kg/m3 and the constant g of gravity is 9.8 m/s2 yן 6m 8m -3m I х frustum of a cone 2 (5 points) Let S be the solid obtained by rotating the region bounded by y = x3, y2 = x about the vertical line y = 2. (a) Using the washer method set up (do not compute!) an integral expressing the volume of solid S. I (b) Using the method of cylindrical shells set up (do not compute!) an integral expressing the volume of solid S. [ 3 (5 points) Find the volume of the solid below by integrating the area of cross-sectional regions. I 2 cm 4 cm 3 cm X 4 (5 points) Find the centroid (x,y) of a lamina (with uniform density) bounded by the x-axis, y-axis and ellipse x2 + 1, pictured below. Note that the area of the lamina here is 3A/2. 4. 9 y2 = V 3 х 2 5 и (5 points) Solve x everyone + y = e-3x with y(1) = 0 for y(x). Hint: Let y and first solve for u. х 6 (5 points) A tank contains 200 L of pure water. Brine that contains 0.08 kg of salt per liter of water enters the tank at a rate of 6 L/min. The solution is kept thoroughly mixed and drains from the tank at a rate of 6 L/min. How much salt is in the tank after one hour? 7 (5 points) The line segment x = 1, y ,0 < y

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