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7.3 Day One: Volumes by Slicing

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3 3 3 Find the volume of the pyramid: Consider a horizontal slice through the pyramid. s dh The volume of the slice is s 2 dh. If we put zero at the top of the pyramid and make down the positive direction, then s=h. 0 3 h This correlates with the formula:

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Method of Slicing: 1 Find a formula for V ( x ). (Note that I used V ( x ) instead of A(x).) Sketch the solid and a typical cross section. 2 3 Find the limits of integration. 4 Integrate V ( x ) to find volume.

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x y A 45 o wedge is cut from a cylinder of radius 3 as shown. Find the volume of the wedge. You could slice this wedge shape several ways, but the simplest cross section is a rectangle. If we let h equal the height of the slice then the volume of the slice is: Since the wedge is cut at a 45 o angle: x h 45 o Since

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x y Even though we started with a cylinder, does not enter the calculation!

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Cavalieri’s Theorem: Two solids with equal altitudes and identical parallel cross sections have the same volume. Identical Cross Sections

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Cavalieri’s Theorem: Volume of a SphereVolume of a Sphere

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7.3 Disk and Washer Methods

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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.

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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: In this case: r= the y value of the function thickness = a small change in x = dx

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The volume of each flat cylinder (disk) is: If we add the volumes, we get:

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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: A shape rotated about the y-axis would be:

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The region between the curve, and the y -axis is revolved about the y -axis. Find the volume. y x We use a horizontal disk. The thickness is dy. The radius is the x value of the function. volume of disk

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The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of this equation revolved about the y-axis: The volume can be calculated using the disk method with a horizontal disk.

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The region bounded by and is revolved about the y-axis. Find the volume. The “disk” now has a hole in it, making it a “washer”. If we use a horizontal slice: The volume of the washer is: outer radius inner radius

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This application of the method of slicing is called the washer method. The shape of the slice is a circle with a hole in it, so we subtract the area of the inner circle from the area of the outer circle. The washer method formula is:

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If the same region is rotated about the line x = 2 : The outer radius is: R The inner radius is: r

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Washer Cross Section The region in the first quadrant enclosed by the y-axis and the graphs of y = cos x and y = sin x is revolved about the x-axis to form a solid. Find its volume.

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Washer Cross Section The region in the first quadrant enclosed by the y-axis and the graphs of y = cos x and y = sin x is revolved about the x-axis to form a solid. Find its volume.

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7.3 The Shell Method

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Find the volume of the region bounded by,, and revolved about the y - axis. We can use the washer method if we split it into two parts: outer radius inner radius thickness of slice cylinder Japanese Spider Crab Georgia Aquarium, Atlanta

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If we take a vertical sliceand revolve it about the y-axis we get a cylinder. cross section If we add all of the cylinders together, we can reconstruct the original object. Here is another way we could approach this problem:

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cross section The volume of a thin, hollow cylinder is given by: r is the x value of the function. h is the y value of the function. thickness is dx.

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cross section If we add all the cylinders from the smallest to the largest: This is called the shell method because we use cylindrical shells.

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Find the volume generated when this shape is revolved about the y axis. We can’t solve for x, so we can’t use a horizontal slice directly.

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Shell method: If we take a vertical slice and revolve it about the y-axis we get a cylinder.

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Note:When entering this into the calculator, be sure to enter the multiplication symbol before the parenthesis.

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When the strip is parallel to the axis of rotation, use the shell method. When the strip is perpendicular to the axis of rotation, use the washer method.

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Find the volume of the solid when the region bounded by the curve y =, the x-axis, and the line x = 4 is revolved about the x-axis. Find the volume of the solid using cylindrical shells.

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Find the volume of the solid of revolution formed by revolving the region bounded by the graph of and the y axis, 0 ≤ y ≤ 1, about the x-axis. Use the Shell Method.

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Find the volume of the solid formed by revolving the region bounded by the graphs y = x 3 + x + 1, y = 1, and x = 1 about the line x = 2.

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