Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 20037.3 day 2Disks, Washers and ShellsLimerick Nuclear Generating Station, Pottstown, Pennsylvania
yx=Suppose I start with this curve.My boss at the ACME Rocket Company has assigned me to build a nose cone in this shape.So I put a piece of wood in a lathe and turn it to a shape to match the curve.→
yx=How could we find the volume of the cone?One way would be to cut it into a series of thin slices (flat cylinders) and add their volumes.The volume of each flat cylinder (disk) is:2the thicknessrπ⋅In this case:r=the yvalue of the functionthickness =a small change in x =dxπ(292xdx→
yx=The volume of each flat cylinder (disk) is:2the thicknessrπ⋅If we add the volumes, we get:(29240xdxπ∫40x dxπ=∫4202xπ=8π=π(292xdx→
This application of the method of slicing is called the disk method. The shape of the slice is a disk, so we use the formula for the area of a circle to find the volume of the disk.If the shape is rotated about the x-axis, then the formula is:2baVydxπ=Since we will be using the disk method to rotate shapes about other lines besides the x-axis, we will not have this formula on the formula quizzes.2baVxdyπ=∫A shape rotated about the y-axis would be:→
The region between the curve , and they-axis is revolved about the y-axis. Find the volume.