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7.2: Volumes by Slicing Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2001 Little Rock Central High School, Little Rock, Arkansas

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The Method of Cross-Sections Intersect S with a plane P x perpendicular to the x-axis Intersect S with a plane P x perpendicular to the x-axis Call the cross- sectional area A(x) Call the cross- sectional area A(x) A(x) will vary as x increases from a to b A(x) will vary as x increases from a to b

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Cross-Sections (cont’d) Divide S into “slabs” of equal width ∆x using planes at x 1, x 2,…, x n Divide S into “slabs” of equal width ∆x using planes at x 1, x 2,…, x n Like slicing a loaf of bread! Like slicing a loaf of bread! To add an infinite number of slices of bread…..we must integrate To add an infinite number of slices of bread…..we must integrate

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The formulacan be applied to any solid for which the cross- sectional area A(x) can be found The formulacan be applied to any solid for which the cross- sectional area A(x) can be found This includes solids of revolution, which we will cover today… This includes solids of revolution, which we will cover today… …but includes many other solids as well …but includes many other solids as well A Bigger Picture

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

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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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Second Example of Washers Problem Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves y = x and y = x 2 Problem Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves y = x and y = x 2 …but about the line y = 2 instead of the x-axis …but about the line y = 2 instead of the x-axis The solid and a cross-section are illustrated on the next slide The solid and a cross-section are illustrated on the next slide

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Second Example of Washers (cont’d)

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Solution Here Solution Here So So

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The formulacan be applied to any solid for which the cross- sectional area A(x) can be found The formulacan be applied to any solid for which the cross- sectional area A(x) can be found This includes solids of revolution, as shown above… This includes solids of revolution, as shown above… …but includes many other solids as well …but includes many other solids as well A Bigger Picture

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The Method of Cross-Sections Intersect S with a plane P x perpendicular to the x-axis Intersect S with a plane P x perpendicular to the x-axis Call the cross- sectional area A(x) Call the cross- sectional area A(x) A(x) will vary as x increases from a to b A(x) will vary as x increases from a to b

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Cross-Sections (cont’d) Divide S into “slabs” of equal width ∆x using planes at x 1, x 2,…, x n Divide S into “slabs” of equal width ∆x using planes at x 1, x 2,…, x n Like slicing a loaf of bread! Like slicing a loaf of bread! To add an infinite number of slices of bread…..we must integrate To add an infinite number of slices of bread…..we must integrate

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